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The Krein–von Neumann extension revisited

2021-06-27
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +3 more
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension … We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
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On domain properties of Bessel-type operators

2022-12-12
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now … Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $ L^2((a, b); dx) $ associated with differential expressions of the form \begin{document}$ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in {\mathbb{R}}, \; x \in (a, b), $\end{document} and$ \begin{align*} \tau_{s_a, s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a, b), & \\ s_a, s_b \in [0, \infty), \; q \in L^{\infty}((a, b); dx), \; q \text{ real-valued a.e. on } (a, b) , & \end{align*} $where $ (a, b) \subset {\mathbb{R}} $ is a bounded interval.As an explicit illustration we describe the Krein–von Neumann extension of the minimal operator corresponding to $ \omega_{s_a} $ and $ \tau_{s_a, s_b} $.
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A Survey of Some Norm Inequalities

2021-01-27
Fritz Gesztesy , Roger Nichols , Jonathan Stanfill
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Bessel-Type Operators and a Refinement of Hardy’s Inequality

2021-01-01
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
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Donoghue 𝑚-functions for Singular Sturm–Liouville operators

2024-04-12
Fritz Gesztesy , Lance L. Littlejohn , R NICHOLS +2 more
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, closed, symmetric operator in the complex, separable Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with equal deficiency indices and denote by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i Baseline equals kernel left-parenthesis left-parenthesis ModifyingAbove upper A With dot right-parenthesis Superscript asterisk Baseline minus i upper I Subscript script upper H Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ker</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}_i = \ker ((\dot {A})^* - i I_{\mathcal {H}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis script upper N Subscript i Baseline right-parenthesis equals k element-of double-struck upper N union StartSet normal infinity EndSet"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\dim (\mathcal {N}_i)=k\in \mathbb {N} \cup \{\infty \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the associated deficiency subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes a self-adjoint extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the Donoghue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper A comma script upper N Sub Subscript i Subscript Superscript upper D o Baseline left-parenthesis dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{A,\mathcal {N}_i}^{Do} (\,\cdot \,)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma script upper N Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,\mathcal {N}_i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper A comma script upper N Sub Subscript i Subscript Superscript upper D o Baseline left-parenthesis z right-parenthesis equals z upper I Subscript script upper N Sub Subscript i Subscript Baseline plus left-parenthesis z squared plus 1 right-parenthesis upper P Subscript script upper N Sub Subscript i Subscript Baseline left-parenthesis upper A minus z upper I Subscript script upper H Baseline right-parenthesis Superscript negative 1 Baseline upper P Subscript script upper N Sub Subscript i Subscript Baseline vertical-bar Subscript script upper N Sub Subscript i Subscript Baseline"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>z</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{A,\mathcal {N}_i}^{Do}(z)=zI_{\mathcal {N}_i} + (z^2+1) P_{\mathcal {N}_i} (A - z I_{\mathcal {H}})^{-1} P_{\mathcal {N}_i} \vert _{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z element-of double-struck upper C minus double-struck upper R comma"> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">z\in \mathbb {C}\setminus \mathbb {R},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript script upper N Sub Subscript i"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">I_{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the identity operator in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript script upper N Sub Subscript i"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">P_{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the orthogonal projection in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assuming the standard local integrability hypotheses on the coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p comma q comma r"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p, q,r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we study all self-adjoint realizations corresponding to the differential expression <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau equals StartFraction 1 Over r left-parenthesis x right-parenthesis EndFraction left-bracket minus StartFraction d Over d x EndFraction p left-parenthesis x right-parenthesis StartFraction d Over d x EndFraction plus q left-parenthesis x right-parenthesis right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau =\frac {1}{r(x)}[-\frac {d}{dx}p(x)\frac {d}{dx} + q(x)]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a.e. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of left-parenthesis a comma b right-parenthesis subset-of-or-equal-to double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">x\in (a,b) \subseteq \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis left-parenthesis a comma b right-parenthesis semicolon r d x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>;</mml:mo> <mml:mi>r</mml:mi> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2((a,b); rdx)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and, as our principal aim in this paper, systematically construct the associated Donoghue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions (respectively, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 times 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>×<!-- × --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2 \times 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices) in all cases where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi>τ<!-- τ --></mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in the limit circle case at least at one interval endpoint <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Spectral $$\varvec{\zeta }$$-functions and $$\varvec{\zeta }$$-regularized functional determinants for regular Sturm–Liouville operators

2021-10-27
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +1 more

Logarithmic refinements of a power weighted Hardy–Rellich-type inequality

2024-08-08
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying … Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $$n \in {\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .
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A refinement of Hardy's inequality

2021-01-29
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where … The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schrodinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$. In addition, we hint at the possibility to extend this inequality to more general situations where the differential expression is of the form $\tau = -(d^2/dx^2) + q(x)$, $x \in (a,b) \subset \mathbb{R}$, where $q(\, \cdot \,)$ behaves like $C_c (x-c)^{-2}$ near $x=c$, $c \in \{a,b\}$, with $C_c\geq - 1/4$.

Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

2020-10-06
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +7 more
Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these … Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.

Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators

2021-01-28
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +1 more
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions … The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $\tau$. Depending on the underlying boundary conditions, we express the $\zeta$-function values in terms of a fundamental system of solutions of $\tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $\zeta$-function through a Liouville transformation and provide an explicit expression for the $\zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrodinger operators with zero, piecewise constant, and a linear potential on a compact interval.

The Jacobi Operator and Its Donoghue m-Functions

2023-01-01
Fritz Gesztesy , Mateusz Piorkowski , Jonathan Stanfill

The Jacobi Operator on $$(-1,1)$$ and Its Various m-Functions

2024-09-17
Fritz Gesztesy , Lance L. Littlejohn , Mateusz Piorkowski +1 more
Abstract We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression $$\begin{aligned} \tau _{\alpha ,\beta } =&amp;- (1-x)^{-\alpha } (1+x)^{-\beta … Abstract We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression $$\begin{aligned} \tau _{\alpha ,\beta } =&amp;- (1-x)^{-\alpha } (1+x)^{-\beta }(d/dx) \big ((1-x)^{\alpha +1}(1+x)^{\beta +1}\big ) (d/dx), \\&amp;\alpha , \beta \in {\mathbb {R}}, \, x \in (-1,1), \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow/> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> in $$L^2\big ((-1,1); (1-x)^{\alpha } (1+x)^{\beta } dx\big )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>;</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>β</mml:mi> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$\alpha , \beta \in {\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> -periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced m -functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.

The spectral $$\zeta $$-function for quasi-regular Sturm–Liouville operators

2025-01-22
Guglielmo Fucci , Mateusz Piorkowski , Jonathan Stanfill
Abstract In this work, we analyze the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function associated with the self-adjoint extensions, $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> … Abstract In this work, we analyze the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function associated with the self-adjoint extensions, $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> , of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> . The characteristic function is then employed to construct a contour integral representation for the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function of $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> . By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function to a larger region of the complex plane. We also present a method for computing the value of the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function of $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s .
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Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

2021-01-11
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +4 more
Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with … Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2, 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.

The Krein-von Neumann extension revisited

2021-02-01
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +3 more
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension … We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.

The Jacobi operator and its Donoghue $m$-functions

2021-01-01
Fritz Gesztesy , Mateusz Piorkowski , Jonathan Stanfill
In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^α (1+x)^β dx\big)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ … In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^α (1+x)^β dx\big)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ 1}(1+x)^{β+ 1}\big) (d/dx),&amp; \\ x \in (-1,1), \; α, β\in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature.
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On domain properties of Bessel-type operators

2021-01-01
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at … Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $L^2((a,b); dx)$ associated with differential expressions of the form \[ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} \tau_{s_a,s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a,b),& \\ s_a, s_b \in [0,\infty), \; q \in L^{\infty}((a,b); dx), \; q \text{ real-valued~a.e.~on $(a,b)$,}& \end{align*} where $(a,b) \subset \mathbb{R}$ is a bounded interval. As an explicit illustration we describe the Krein-von Neumann extension of the minimal operator corresponding $\omega_{s_a}$ and $\tau_{s_a,s_b}$.

Donoghue $m$-functions for singular Sturm--Liouville operators

2021-01-01
Fritz Gesztesy , Lance L. Littlejohn , Roger Nichols +2 more
Let $\dot A$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by $\mathcal{N}_i = \ker \big(\big(\dot A\big)^* - … Let $\dot A$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by $\mathcal{N}_i = \ker \big(\big(\dot A\big)^* - i I_{\mathcal{H}}\big)$, $\dim \, (\mathcal{N}_i)=k\in \mathbb{N} \cup \{\infty\}$, the associated deficiency subspace of $\dot A$ . If $A$ denotes a self-adjoint extension of $\dot A$ in $\mathcal{H}$, the Donoghue $m$-operator $M_{A,\mathcal{N}_i}^{Do} (\, \cdot \,)$ in $\mathcal{N}_i$ associated with the pair $(A,\mathcal{N}_i)$ is given by \[ M_{A,\mathcal{N}_i}^{Do}(z)=zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A - z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i} \big\vert_{\mathcal{N}_i}\,, \quad z\in \mathbb{C} \backslash \mathbb{R}, \] with $I_{\mathcal{N}_i}$ the identity operator in $\mathcal{N}_i$, and $P_{\mathcal{N}_i}$ the orthogonal projection in $\mathcal{H}$ onto $\mathcal{N}_i$. Assuming the standard local integrability hypotheses on the coefficients $p, q,r$, we study all self-adjoint realizations corresponding to the differential expression \[ \tau=\frac{1}{r(x)}\left[-\frac{d}{dx}p(x)\frac{d}{dx} + q(x)\right] \, \text{ for a.e. $x\in(a,b) \subseteq \mathbb{R}$,} \] in $L^2((a,b); rdx)$, and, as the principal aim of this paper, systematically construct the associated Donoghue $m$-functions (resp., $2 \times 2$ matrices) in all cases where $\tau$ is in the limit circle case at least at one interval endpoint $a$ or $b$.
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The Krein-von Neumann extension revisited

2021-01-01
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +3 more
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension … We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
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Bessel-Type Operators and a refinement of Hardy's inequality

2021-01-01
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in … The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schr\"{o}dinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$.
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Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators

2021-01-01
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +1 more
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions … The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $\tau$. Depending on the underlying boundary conditions, we express the $\zeta$-function values in terms of a fundamental system of solutions of $\tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $\zeta$-function through a Liouville transformation and provide an explicit expression for the $\zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schr\"{o}dinger operators with zero, piecewise constant, and a linear potential on a compact interval.
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Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

2020-01-01
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +7 more
Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these … Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.
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Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

2021-01-01
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +4 more
Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with … Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2,\ 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.
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Partitions with Designated Summands Not Divisible by \(2^\ell\), 2 and \(3^\ell\) Modulo 2, 4, and 3

2023-06-12
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +4 more
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The Jacobi operator on $(-1,1)$ and its various $m$-functions

2023-01-01
Fritz Gesztesy , Lance L. Littlejohn , Mateusz Piorkowski +1 more
We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\ \alpha, … We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\ \alpha, \beta \in \mathbb{R}, \; x \in (-1,1),& \end{align*} in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, $\alpha, \beta \in \mathbb{R}$. In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general $\eta$-periodic and Krein--von Neumann extensions. In particular, we treat all underlying Weyl-Titchmarsh-Kodaira and Green's function induced $m$-functions and revisit their Nevanlinna-Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.
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Logarithmic Refinements of a Power Weighted Hardy--Rellich-Type Inequality

2024-07-06
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and … The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $n \in \mathbb{N}$, $n\geq 2$.
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Finer limit circle/limit point classification for Sturm-Liouville operators

2024-07-05
Mateusz Piorkowski , Jonathan Stanfill
In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace \infty \rbrace$ which we call the `regularization index' associated to the endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential expressions … In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace \infty \rbrace$ which we call the `regularization index' associated to the endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential expressions with trace class resolvents. This notion extends the limit circle/limit point dichotomy in the sense that $\ell_c~=~0$ at some endpoint if and only if the expression is in the limit circle case. In the limit point case $\ell_c>0$, a natural interpretation in terms of iterated Darboux transforms is provided. We also show stability of the index $\ell_c$ for a suitable class of perturbations, extending earlier work on perturbations of spherical Schr\"odinger operators to the case of general three-coefficient Sturm-Liouville operators. We demonstrate our results by considering a variety of examples including generalized Bessel operators, Jacobi differential operators, and Schr\"odinger operators on the half-line with power potentials.
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Factorizations and Power Weighted Rellich and Hardy--Rellich-Type Inequalities

2024-07-26
Fritz Gesztesy , Michael M. H. Pang , Jake Parmentier +1 more
We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke. We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke.
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Nonnegative extensions of Sturm-Liouville operators with an application to problems with symmetric coefficient functions

2024-10-06
Christoph Fischbacher , Jonathan Stanfill
The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm-Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative … The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm-Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative self-adjoint extensions of the minimal operator in terms of generalized boundary values, as well as a parameterization of all nonnegative extensions when fixing a boundary condition at one endpoint. In addition, we investigate problems where the coefficient functions are symmetric about the midpoint of a finite interval, illustrating how every self-adjoint operator of this form is unitarily equivalent to the direct sum of two self-adjoint operators restricted to half of the interval. We also extend these result to symmetric two interval problems. We then apply our previous results to parameterize all nonnegative extensions of operators with symmetric coefficient functions. We end with an example of an operator with a symmetric Bessel-type potential (i.e., symmetric confining potential) and an application to integral inequalities.
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The spectral $\zeta$-function for quasi-regular Sturm--Liouville operators

2024-09-10
Guglielmo Fucci , Mateusz Piorkowski , Jonathan Stanfill
In this work we analyze the spectral $\zeta$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we … In this work we analyze the spectral $\zeta$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we find the characteristic function which implicitly provides the eigenvalues associated with a given self-adjoint extension $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $\zeta$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $\zeta$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $\zeta$-function of $T_{A,B}$ at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $\zeta$-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of $s$.
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Correction to: Logarithmic refinements of a power weighted Hardy–Rellich-type inequality

2024-11-19
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill

The exotic structure of the spectral $\zeta$-function for the Schr\"odinger operator with P\"oschl--Teller potential

2024-11-26
Guglielmo Fucci , Jonathan Stanfill
This work focuses on the analysis of the spectral $\zeta$-function associated with a Schr\"{o}dinger operator endowed with a P\"oschl--Teller potential. We construct the spectral $\zeta$-function using a contour integral representation … This work focuses on the analysis of the spectral $\zeta$-function associated with a Schr\"{o}dinger operator endowed with a P\"oschl--Teller potential. We construct the spectral $\zeta$-function using a contour integral representation and, for particular self-adjoint extensions, we perform its analytic continuation to a larger region of the complex plane. We show that the spectral $\zeta$-function in these cases can possess a very unusual and remarkable structure consisting of a series of logarithmic branch points located at every nonpositive integer value of $s$ along with infinitely many additional branch points (and finitely many simple poles) whose locations depend on the parameters of the problem. By comparing the P\"oschl--Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by a smooth potential on a finite interval can greatly affect the meromorphic structure and branch points of the spectral $\zeta$-function in surprising ways.
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Nonnegative extensions of Sturm–Liouville operators with an application to problems with symmetric coefficient functions

2025-02-05
Christoph Fischbacher , Jonathan Stanfill

Factorizations and Power Weighted Rellich and Hardy–Rellich-Type Inequalities

2025-04-24
Fritz Gesztesy , Michael M. H. Pang , Jake Parmentier +1 more

Factorizations and Power Weighted Rellich and Hardy–Rellich-Type Inequalities

2025-04-24
Fritz Gesztesy , Michael M. H. Pang , Jake Parmentier +1 more

Nonnegative extensions of Sturm–Liouville operators with an application to problems with symmetric coefficient functions

2025-02-05
Christoph Fischbacher , Jonathan Stanfill

The spectral $$\zeta $$-function for quasi-regular Sturm–Liouville operators

2025-01-22
Guglielmo Fucci , Mateusz Piorkowski , Jonathan Stanfill
Abstract In this work, we analyze the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function associated with the self-adjoint extensions, $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> … Abstract In this work, we analyze the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function associated with the self-adjoint extensions, $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> , of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> . The characteristic function is then employed to construct a contour integral representation for the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function of $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> . By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function to a larger region of the complex plane. We also present a method for computing the value of the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function of $$T_{A,B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> </mml:math> at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s .
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The exotic structure of the spectral $\zeta$-function for the Schr\"odinger operator with P\"oschl--Teller potential

2024-11-26
Guglielmo Fucci , Jonathan Stanfill
This work focuses on the analysis of the spectral $\zeta$-function associated with a Schr\"{o}dinger operator endowed with a P\"oschl--Teller potential. We construct the spectral $\zeta$-function using a contour integral representation … This work focuses on the analysis of the spectral $\zeta$-function associated with a Schr\"{o}dinger operator endowed with a P\"oschl--Teller potential. We construct the spectral $\zeta$-function using a contour integral representation and, for particular self-adjoint extensions, we perform its analytic continuation to a larger region of the complex plane. We show that the spectral $\zeta$-function in these cases can possess a very unusual and remarkable structure consisting of a series of logarithmic branch points located at every nonpositive integer value of $s$ along with infinitely many additional branch points (and finitely many simple poles) whose locations depend on the parameters of the problem. By comparing the P\"oschl--Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by a smooth potential on a finite interval can greatly affect the meromorphic structure and branch points of the spectral $\zeta$-function in surprising ways.
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Correction to: Logarithmic refinements of a power weighted Hardy–Rellich-type inequality

2024-11-19
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill

Nonnegative extensions of Sturm-Liouville operators with an application to problems with symmetric coefficient functions

2024-10-06
Christoph Fischbacher , Jonathan Stanfill
The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm-Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative … The purpose of this paper is to study nonnegative self-adjoint extensions associated with singular Sturm-Liouville expressions with strictly positive minimal operators. We provide a full characterization of all possible nonnegative self-adjoint extensions of the minimal operator in terms of generalized boundary values, as well as a parameterization of all nonnegative extensions when fixing a boundary condition at one endpoint. In addition, we investigate problems where the coefficient functions are symmetric about the midpoint of a finite interval, illustrating how every self-adjoint operator of this form is unitarily equivalent to the direct sum of two self-adjoint operators restricted to half of the interval. We also extend these result to symmetric two interval problems. We then apply our previous results to parameterize all nonnegative extensions of operators with symmetric coefficient functions. We end with an example of an operator with a symmetric Bessel-type potential (i.e., symmetric confining potential) and an application to integral inequalities.
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The Jacobi Operator on $$(-1,1)$$ and Its Various m-Functions

2024-09-17
Fritz Gesztesy , Lance L. Littlejohn , Mateusz Piorkowski +1 more
Abstract We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression $$\begin{aligned} \tau _{\alpha ,\beta } =&amp;- (1-x)^{-\alpha } (1+x)^{-\beta … Abstract We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression $$\begin{aligned} \tau _{\alpha ,\beta } =&amp;- (1-x)^{-\alpha } (1+x)^{-\beta }(d/dx) \big ((1-x)^{\alpha +1}(1+x)^{\beta +1}\big ) (d/dx), \\&amp;\alpha , \beta \in {\mathbb {R}}, \, x \in (-1,1), \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow/> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> in $$L^2\big ((-1,1); (1-x)^{\alpha } (1+x)^{\beta } dx\big )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>;</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>β</mml:mi> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$\alpha , \beta \in {\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> -periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced m -functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.

The spectral $\zeta$-function for quasi-regular Sturm--Liouville operators

2024-09-10
Guglielmo Fucci , Mateusz Piorkowski , Jonathan Stanfill
In this work we analyze the spectral $\zeta$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we … In this work we analyze the spectral $\zeta$-function associated with the self-adjoint extensions, $T_{A,B}$, of quasi-regular Sturm--Liouville operators that are bounded from below. By utilizing the Green's function formalism, we find the characteristic function which implicitly provides the eigenvalues associated with a given self-adjoint extension $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $\zeta$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $\zeta$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $\zeta$-function of $T_{A,B}$ at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral $\zeta$-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of $s$.
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Logarithmic refinements of a power weighted Hardy–Rellich-type inequality

2024-08-08
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying … Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $$n \in {\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .
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Factorizations and Power Weighted Rellich and Hardy--Rellich-Type Inequalities

2024-07-26
Fritz Gesztesy , Michael M. H. Pang , Jake Parmentier +1 more
We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke. We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke.
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Logarithmic Refinements of a Power Weighted Hardy--Rellich-Type Inequality

2024-07-06
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and … The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $n \in \mathbb{N}$, $n\geq 2$.
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Finer limit circle/limit point classification for Sturm-Liouville operators

2024-07-05
Mateusz Piorkowski , Jonathan Stanfill
In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace \infty \rbrace$ which we call the `regularization index' associated to the endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential expressions … In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace \infty \rbrace$ which we call the `regularization index' associated to the endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential expressions with trace class resolvents. This notion extends the limit circle/limit point dichotomy in the sense that $\ell_c~=~0$ at some endpoint if and only if the expression is in the limit circle case. In the limit point case $\ell_c>0$, a natural interpretation in terms of iterated Darboux transforms is provided. We also show stability of the index $\ell_c$ for a suitable class of perturbations, extending earlier work on perturbations of spherical Schr\"odinger operators to the case of general three-coefficient Sturm-Liouville operators. We demonstrate our results by considering a variety of examples including generalized Bessel operators, Jacobi differential operators, and Schr\"odinger operators on the half-line with power potentials.
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Donoghue 𝑚-functions for Singular Sturm–Liouville operators

2024-04-12
Fritz Gesztesy , Lance L. Littlejohn , R NICHOLS +2 more
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, closed, symmetric operator in the complex, separable Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with equal deficiency indices and denote by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i Baseline equals kernel left-parenthesis left-parenthesis ModifyingAbove upper A With dot right-parenthesis Superscript asterisk Baseline minus i upper I Subscript script upper H Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ker</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}_i = \ker ((\dot {A})^* - i I_{\mathcal {H}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis script upper N Subscript i Baseline right-parenthesis equals k element-of double-struck upper N union StartSet normal infinity EndSet"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\dim (\mathcal {N}_i)=k\in \mathbb {N} \cup \{\infty \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the associated deficiency subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes a self-adjoint extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the Donoghue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper A comma script upper N Sub Subscript i Subscript Superscript upper D o Baseline left-parenthesis dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{A,\mathcal {N}_i}^{Do} (\,\cdot \,)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma script upper N Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,\mathcal {N}_i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper A comma script upper N Sub Subscript i Subscript Superscript upper D o Baseline left-parenthesis z right-parenthesis equals z upper I Subscript script upper N Sub Subscript i Subscript Baseline plus left-parenthesis z squared plus 1 right-parenthesis upper P Subscript script upper N Sub Subscript i Subscript Baseline left-parenthesis upper A minus z upper I Subscript script upper H Baseline right-parenthesis Superscript negative 1 Baseline upper P Subscript script upper N Sub Subscript i Subscript Baseline vertical-bar Subscript script upper N Sub Subscript i Subscript Baseline"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>z</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{A,\mathcal {N}_i}^{Do}(z)=zI_{\mathcal {N}_i} + (z^2+1) P_{\mathcal {N}_i} (A - z I_{\mathcal {H}})^{-1} P_{\mathcal {N}_i} \vert _{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z element-of double-struck upper C minus double-struck upper R comma"> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">z\in \mathbb {C}\setminus \mathbb {R},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript script upper N Sub Subscript i"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">I_{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the identity operator in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript script upper N Sub Subscript i"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">P_{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the orthogonal projection in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assuming the standard local integrability hypotheses on the coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p comma q comma r"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p, q,r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we study all self-adjoint realizations corresponding to the differential expression <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau equals StartFraction 1 Over r left-parenthesis x right-parenthesis EndFraction left-bracket minus StartFraction d Over d x EndFraction p left-parenthesis x right-parenthesis StartFraction d Over d x EndFraction plus q left-parenthesis x right-parenthesis right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau =\frac {1}{r(x)}[-\frac {d}{dx}p(x)\frac {d}{dx} + q(x)]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a.e. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of left-parenthesis a comma b right-parenthesis subset-of-or-equal-to double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">x\in (a,b) \subseteq \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis left-parenthesis a comma b right-parenthesis semicolon r d x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>;</mml:mo> <mml:mi>r</mml:mi> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2((a,b); rdx)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and, as our principal aim in this paper, systematically construct the associated Donoghue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions (respectively, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 times 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>×<!-- × --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2 \times 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices) in all cases where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi>τ<!-- τ --></mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in the limit circle case at least at one interval endpoint <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Partitions with Designated Summands Not Divisible by \(2^\ell\), 2 and \(3^\ell\) Modulo 2, 4, and 3

2023-06-12
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +4 more
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The Jacobi Operator and Its Donoghue m-Functions

2023-01-01
Fritz Gesztesy , Mateusz Piorkowski , Jonathan Stanfill

The Jacobi operator on $(-1,1)$ and its various $m$-functions

2023-01-01
Fritz Gesztesy , Lance L. Littlejohn , Mateusz Piorkowski +1 more
We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\ \alpha, … We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\ \alpha, \beta \in \mathbb{R}, \; x \in (-1,1),& \end{align*} in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, $\alpha, \beta \in \mathbb{R}$. In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general $\eta$-periodic and Krein--von Neumann extensions. In particular, we treat all underlying Weyl-Titchmarsh-Kodaira and Green's function induced $m$-functions and revisit their Nevanlinna-Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.
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On domain properties of Bessel-type operators

2022-12-12
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now … Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $ L^2((a, b); dx) $ associated with differential expressions of the form \begin{document}$ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in {\mathbb{R}}, \; x \in (a, b), $\end{document} and$ \begin{align*} \tau_{s_a, s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a, b), & \\ s_a, s_b \in [0, \infty), \; q \in L^{\infty}((a, b); dx), \; q \text{ real-valued a.e. on } (a, b) , & \end{align*} $where $ (a, b) \subset {\mathbb{R}} $ is a bounded interval.As an explicit illustration we describe the Krein–von Neumann extension of the minimal operator corresponding to $ \omega_{s_a} $ and $ \tau_{s_a, s_b} $.
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Spectral $$\varvec{\zeta }$$-functions and $$\varvec{\zeta }$$-regularized functional determinants for regular Sturm–Liouville operators

2021-10-27
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +1 more

The Krein–von Neumann extension revisited

2021-06-27
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +3 more
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension … We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
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The Krein-von Neumann extension revisited

2021-02-01
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +3 more
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension … We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.

A refinement of Hardy's inequality

2021-01-29
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where … The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schrodinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$. In addition, we hint at the possibility to extend this inequality to more general situations where the differential expression is of the form $\tau = -(d^2/dx^2) + q(x)$, $x \in (a,b) \subset \mathbb{R}$, where $q(\, \cdot \,)$ behaves like $C_c (x-c)^{-2}$ near $x=c$, $c \in \{a,b\}$, with $C_c\geq - 1/4$.

Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators

2021-01-28
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +1 more
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions … The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $\tau$. Depending on the underlying boundary conditions, we express the $\zeta$-function values in terms of a fundamental system of solutions of $\tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $\zeta$-function through a Liouville transformation and provide an explicit expression for the $\zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrodinger operators with zero, piecewise constant, and a linear potential on a compact interval.

A Survey of Some Norm Inequalities

2021-01-27
Fritz Gesztesy , Roger Nichols , Jonathan Stanfill
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Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

2021-01-11
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +4 more
Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with … Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2, 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.

Bessel-Type Operators and a Refinement of Hardy’s Inequality

2021-01-01
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
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The Jacobi operator and its Donoghue $m$-functions

2021-01-01
Fritz Gesztesy , Mateusz Piorkowski , Jonathan Stanfill
In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^α (1+x)^β dx\big)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ … In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^α (1+x)^β dx\big)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ 1}(1+x)^{β+ 1}\big) (d/dx),&amp; \\ x \in (-1,1), \; α, β\in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature.
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On domain properties of Bessel-type operators

2021-01-01
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at … Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $L^2((a,b); dx)$ associated with differential expressions of the form \[ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} \tau_{s_a,s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a,b),& \\ s_a, s_b \in [0,\infty), \; q \in L^{\infty}((a,b); dx), \; q \text{ real-valued~a.e.~on $(a,b)$,}& \end{align*} where $(a,b) \subset \mathbb{R}$ is a bounded interval. As an explicit illustration we describe the Krein-von Neumann extension of the minimal operator corresponding $\omega_{s_a}$ and $\tau_{s_a,s_b}$.

Donoghue $m$-functions for singular Sturm--Liouville operators

2021-01-01
Fritz Gesztesy , Lance L. Littlejohn , Roger Nichols +2 more
Let $\dot A$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by $\mathcal{N}_i = \ker \big(\big(\dot A\big)^* - … Let $\dot A$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by $\mathcal{N}_i = \ker \big(\big(\dot A\big)^* - i I_{\mathcal{H}}\big)$, $\dim \, (\mathcal{N}_i)=k\in \mathbb{N} \cup \{\infty\}$, the associated deficiency subspace of $\dot A$ . If $A$ denotes a self-adjoint extension of $\dot A$ in $\mathcal{H}$, the Donoghue $m$-operator $M_{A,\mathcal{N}_i}^{Do} (\, \cdot \,)$ in $\mathcal{N}_i$ associated with the pair $(A,\mathcal{N}_i)$ is given by \[ M_{A,\mathcal{N}_i}^{Do}(z)=zI_{\mathcal{N}_i} + (z^2+1) P_{\mathcal{N}_i} (A - z I_{\mathcal{H}})^{-1} P_{\mathcal{N}_i} \big\vert_{\mathcal{N}_i}\,, \quad z\in \mathbb{C} \backslash \mathbb{R}, \] with $I_{\mathcal{N}_i}$ the identity operator in $\mathcal{N}_i$, and $P_{\mathcal{N}_i}$ the orthogonal projection in $\mathcal{H}$ onto $\mathcal{N}_i$. Assuming the standard local integrability hypotheses on the coefficients $p, q,r$, we study all self-adjoint realizations corresponding to the differential expression \[ \tau=\frac{1}{r(x)}\left[-\frac{d}{dx}p(x)\frac{d}{dx} + q(x)\right] \, \text{ for a.e. $x\in(a,b) \subseteq \mathbb{R}$,} \] in $L^2((a,b); rdx)$, and, as the principal aim of this paper, systematically construct the associated Donoghue $m$-functions (resp., $2 \times 2$ matrices) in all cases where $\tau$ is in the limit circle case at least at one interval endpoint $a$ or $b$.
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The Krein-von Neumann extension revisited

2021-01-01
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +3 more
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension … We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
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Bessel-Type Operators and a refinement of Hardy's inequality

2021-01-01
Fritz Gesztesy , Michael M. H. Pang , Jonathan Stanfill
The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in … The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schr\"{o}dinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$.
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Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators

2021-01-01
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +1 more
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions … The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $\zeta$-functions to efficiently compute values of spectral $\zeta$-functions at positive integers associated to regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions $\tau$. Depending on the underlying boundary conditions, we express the $\zeta$-function values in terms of a fundamental system of solutions of $\tau y = z y$ and their expansions about the spectral point $z=0$. Furthermore, we give the full analytic continuation of the $\zeta$-function through a Liouville transformation and provide an explicit expression for the $\zeta$-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schr\"{o}dinger operators with zero, piecewise constant, and a linear potential on a compact interval.
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Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$

2021-01-01
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +4 more
Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with … Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2,\ 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.
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Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

2020-10-06
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +7 more
Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these … Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.

Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

2020-01-01
Daniel Herden , Mark R. Sepanski , Jonathan Stanfill +7 more
Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these … Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.
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Coauthor Papers Together
Fritz Gesztesy 23
Michael M. H. Pang 10
Guglielmo Fucci 9
Mateusz Piorkowski 8
Lance L. Littlejohn 7
Klaus Kirsten 6
Mark R. Sepanski 5
Daniel Herden 5
Joel Henningsen 5
Cordell Hammon 5
Roger Nichols 3
Henry Ickes 3
Indalecio Ruiz 3
Indalecio Ruiz 2
Christoph Fischbacher 2
Taylor Poe 2
Jorge Marchena Menendez 2
Henry Ickes 2
Edward L. Smith 2
Jake Parmentier 2
Roger Nichols 2
Michał Piórkowski 1
R NICHOLS 1

Singular Sturm-Liouville Problems: The Friedrichs Extension and Comparison of Eigenvalues

1992-05-01
Heinz‐Dieter Niessen , Anton Zettl
A new characterization of singular self-adjoint boundary conditions for Sturm-Liouville problems is given. These are an exact parallel of the regular case. They are given explicitly in terms of principal … A new characterization of singular self-adjoint boundary conditions for Sturm-Liouville problems is given. These are an exact parallel of the regular case. They are given explicitly in terms of principal and non-principal solutions. The special nature of the Friedrichs extension is clearly apparent and highlighted. Inequalities among the eigenvalues of different boundary conditions, separated and coupled, are obtained. Most of all we want to stress the method of proof. It is based on a very elementary transformation which transforms any singular non-oscillatory limit-circle endpoint into a regular one.

On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from below

2020-06-07
Fritz Gesztesy , Lance L. Littlejohn , Roger Nichols
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A Characterization of the Friedrichs Extension of Sturm-Liouville Operators

1978-06-01
Hubert Kalf

A New Characterization of the Friedrichs Extension of Semibounded Sturm-Liouville Operators

1985-06-01
R. F. Rosenberger

Halbbeschr�nkte gew�hnliche Differentialoperatoren zweiter Ordnung

1950-10-01
Franz Rellich

Eigenwerttheorie gewöhnlicher Differentialgleichungen

1976-01-01
Konrad Jörgens , Franz Rellich

Linear Operators in Hilbert Spaces

1980-01-01
Joachim Weidmann

The Krein–von Neumann extension revisited

2021-06-27
Guglielmo Fucci , Fritz Gesztesy , Klaus Kirsten +3 more
We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension … We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.
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Die zul�ssigen Randbedingrungen bei den singul�ren Eingenwertproblemen der mathematischen Physik

1943-12-01
Franz Rellich

Handbook of Mathematical Functions

1966-02-01
Donald A. McQuarrie
First Page First Page

Linear operators. Part II. Spectral theory

2003-01-01
Nelson Dunford , Jacob T. Schwartz

On the Assignment of Asymptotic Values for the Solutions of Linear Differential Equations of Second Order

1955-07-01
Philip Hartman , Aurel Wintner
Suppose that, corresponding to every given value, c, the diffeveiitial e(uation (1) has a unique solutioni x =-.r (t) satisfying (C ) xc a s t ->oo (so that x( … Suppose that, corresponding to every given value, c, the diffeveiitial e(uation (1) has a unique solutioni x =-.r (t) satisfying (C ) xc a s t ->oo (so that x( oX) can be assigned as an initial condition and this siiigle integration constant determines the solution x(t) uniquely, even though4 (1) is of second order). Then (1) will be called of type (#:). Clearly, (1) i< of type (*) if and only if (1) has exactly one solution satisfyilng the case c = 1 of (2) and, in addition, the case c = 0 of (2) is satisfied only by the triivial solution ( 0). it is also clear that, silnce the general solution of (1) conitains two arbitrary constants, x( oo) calnlot exist (as a finite limit) fom all solutioins x(t), if (1) is of type (*). If x(oo) exists (as a finite limit) for all solutiolns and if x(0o) 7 0 holds for some solution, then (1) will be called of type (:. Clearly, this will be the case if alnd oinly if there exist a solution satisfying the case c 1 of (2) alnd a lnoln-trivial solution ( 0) satisfying the case c= 0 of (2). By adding to the former any constalnt multiple of the latter, it is seell that (1) is of tAype (**) if alnd oinly if there belolngs to some and/or every c:7 0 two, linearly indepelndelnt, solutiolns x(t) satisfying (2). The following theorem will first be proved: (I) In order that (1) be of type (: ), it is sufficient that (i) the coefficient functions of (1) satisfy the conditions

Boundary Value Problems, Weyl Functions, and Differential Operators

2020-01-01
Jussi Behrndt , Seppo Hassi , Henk de Snoo
This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory. It may serve as a reference text for researchers in the … This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory. It may serve as a reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory.
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Singular quadratic functionals

1936-01-01
Marston Morse , Walter Leighton
Introduction.Singular quadratic functions of the type (1.1) have been briefly investigated by Kemble [l ] in connection with Quantum Mechanics.The results of Kemble are of a relatively restricted character.On the … Introduction.Singular quadratic functions of the type (1.1) have been briefly investigated by Kemble [l ] in connection with Quantum Mechanics.The results of Kemble are of a relatively restricted character.On the other hand, Hardy, Littlewood, and Pólya,f in Chapter VII of [1 ] have studied special examples of these functionals employing special methods.The developments of this paper apparently give a systematic approach to the problem of minimizing singular quadratic functionals of the type (1.1).In particular, the results obtained include and generalize Theorems ( 254) and ( 253) of H.L.P. See Examples 9.2 and 12.1 of this paper.The authors have admitted various classes of comparison curves.The results show in striking fashion how the existence of the minimum depends upon the classes of curves admitted.The problem requires a remodeling of the conjugate point theory and an introduction of a new condition called the singularity condition.Lebesgue integrals or their extensions are used throughout.The results of this paper will be applied to extend the theory of characteristic roots and solutions of the related boundary problems.I.
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A Catalogue of Sturm-Liouville Differential Equations

2005-07-20
W. N. Everitt
This catalogue commences with sections devoted to a brief summary of Sturm-Liouville theory including some details of differential expressions and equations, Hilbert function spaces, differential operators, classification of interval endpoints, … This catalogue commences with sections devoted to a brief summary of Sturm-Liouville theory including some details of differential expressions and equations, Hilbert function spaces, differential operators, classification of interval endpoints, boundary condition functions and the Liouville transform. There follows a collection of more than 50 examples of Sturm-Liouville differential equations; many of these examples are connected with well-known special functions, and with problems in mathematical physics and applied mathematics. For most of these examples the interval endpoints are classified within the relevant Hilbert function space, and boundary condition functions are given to determine the domains of the relevant differential operators. In many cases the spectra of these operators are given. The author is indebted to many colleagues who have responded to requests for examples and who checked successive drafts of the catalogue.

Linear differential operators

1967-01-01
M. A. Naĭmark

Donoghue 𝑚-functions for Singular Sturm–Liouville operators

2024-04-12
Fritz Gesztesy , Lance L. Littlejohn , R NICHOLS +2 more
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a densely defined, closed, symmetric operator in the complex, separable Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with equal deficiency indices and denote by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i Baseline equals kernel left-parenthesis left-parenthesis ModifyingAbove upper A With dot right-parenthesis Superscript asterisk Baseline minus i upper I Subscript script upper H Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ker</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {N}_i = \ker ((\dot {A})^* - i I_{\mathcal {H}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis script upper N Subscript i Baseline right-parenthesis equals k element-of double-struck upper N union StartSet normal infinity EndSet"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\dim (\mathcal {N}_i)=k\in \mathbb {N} \cup \{\infty \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the associated deficiency subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes a self-adjoint extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper A With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>A</mml:mi> <mml:mo>˙<!-- ˙ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the Donoghue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper A comma script upper N Sub Subscript i Subscript Superscript upper D o Baseline left-parenthesis dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{A,\mathcal {N}_i}^{Do} (\,\cdot \,)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with the pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma script upper N Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,\mathcal {N}_i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper A comma script upper N Sub Subscript i Subscript Superscript upper D o Baseline left-parenthesis z right-parenthesis equals z upper I Subscript script upper N Sub Subscript i Subscript Baseline plus left-parenthesis z squared plus 1 right-parenthesis upper P Subscript script upper N Sub Subscript i Subscript Baseline left-parenthesis upper A minus z upper I Subscript script upper H Baseline right-parenthesis Superscript negative 1 Baseline upper P Subscript script upper N Sub Subscript i Subscript Baseline vertical-bar Subscript script upper N Sub Subscript i Subscript Baseline"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>D</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>z</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{A,\mathcal {N}_i}^{Do}(z)=zI_{\mathcal {N}_i} + (z^2+1) P_{\mathcal {N}_i} (A - z I_{\mathcal {H}})^{-1} P_{\mathcal {N}_i} \vert _{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z element-of double-struck upper C minus double-struck upper R comma"> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo class="MJX-variant">∖<!-- ∖ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">z\in \mathbb {C}\setminus \mathbb {R},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript script upper N Sub Subscript i"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">I_{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the identity operator in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript script upper N Sub Subscript i"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">P_{\mathcal {N}_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the orthogonal projection in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">N</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {N}_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assuming the standard local integrability hypotheses on the coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p comma q comma r"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p, q,r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we study all self-adjoint realizations corresponding to the differential expression <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau equals StartFraction 1 Over r left-parenthesis x right-parenthesis EndFraction left-bracket minus StartFraction d Over d x EndFraction p left-parenthesis x right-parenthesis StartFraction d Over d x EndFraction plus q left-parenthesis x right-parenthesis right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau =\frac {1}{r(x)}[-\frac {d}{dx}p(x)\frac {d}{dx} + q(x)]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a.e. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of left-parenthesis a comma b right-parenthesis subset-of-or-equal-to double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">x\in (a,b) \subseteq \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis left-parenthesis a comma b right-parenthesis semicolon r d x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>;</mml:mo> <mml:mi>r</mml:mi> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2((a,b); rdx)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and, as our principal aim in this paper, systematically construct the associated Donoghue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions (respectively, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 times 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>×<!-- × --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2 \times 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices) in all cases where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi>τ<!-- τ --></mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in the limit circle case at least at one interval endpoint <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Spectral theory for perturbed Krein Laplacians in nonsmooth domains

2009-10-28
Mark S. Ashbaugh , Fritz Gesztesy , Marius Mitrea +1 more
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On the characterization of the Friedrichs extension of ordinary or elliptic differential operators with a strongly singular potential

1972-06-01
Hubert Kalf

Principal Solutions Revisited

2016-01-01
Stephen Clark , Fritz Gesztesy , Roger Nichols
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Lineare Operatoren in Hilberträumen

2000-01-01
Joachim Weidmann
Seit Erscheinen meines Buches "Lineare Operatoren in Hilberträumen" [38] im Jahre 1976 und dessen englischer Übersetzung [39] im Jahre 1980 haben mich viele freundliche Stellungnahmen erreicht. Häufig Seit Erscheinen meines Buches "Lineare Operatoren in Hilberträumen" [38] im Jahre 1976 und dessen englischer Übersetzung [39] im Jahre 1980 haben mich viele freundliche Stellungnahmen erreicht. Häufig

Boundary triples and Weyl m-functions for powers of the Jacobi differential operator

2020-06-16
Dale Frymark
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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator

2018-06-18
Tom H. Koornwinder , Aleksey Kostenko , Gerald Teschl
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Jacobi–Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression

2006-12-09
W. N. Everitt , K.H. Kwon , Lance L. Littlejohn +2 more

Boundary data maps and Krein's resolvent formula for Sturm-Liouville operators on a finite interval

2014-01-01
Stephen Clark , Fritz Gesztesy , Roger Nichols +1 more
We continue the study of boundary data maps, that is, generalizations of spectral parameter dependent Dirichlet-to-Neumann maps for (three-coefficient) Sturm-Liouville operators on the finite interval (a,b) , to more general … We continue the study of boundary data maps, that is, generalizations of spectral parameter dependent Dirichlet-to-Neumann maps for (three-coefficient) Sturm-Liouville operators on the finite interval (a,b) , to more general boundary conditions, began in [8] and [17].While these earlier studies of boundary data maps focused on the case of general separated boundary conditions at a and b , the present work develops a unified treatment for all possible self-adjoint boundary conditions (i.e., separated as well as non-separated ones).In the course of this paper we describe the connections with Krein's resolvent formula for self-adjoint extensions of the underlying minimal Sturm-Liouville operator (parametrized in terms of boundary conditions), with some emphasis on the Krein extension, develop the basic trace formulas for resolvent differences of self-adjoint extensions, especially, in terms of the associated spectral shift functions, and describe the connections between various parametrizations of all self-adjoint extensions, including the precise relation to von Neumann's basic parametrization in terms of unitary maps between deficiency subspaces.
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Characterizations of the Friedrichs Extensions of Singular Sturm–Liouville Expressions

1986-07-01
Hans G. Kaper , Man Kam Kwong , Anton Zettl
Previous article Next article Characterizations of the Friedrichs Extensions of Singular Sturm–Liouville ExpressionsHans G. Kaper, Man Kam Kwong, and Anton ZettlHans G. Kaper, Man Kam Kwong, and Anton Zettlhttps://doi.org/10.1137/0517056PDFBibTexSections ToolsAdd … Previous article Next article Characterizations of the Friedrichs Extensions of Singular Sturm–Liouville ExpressionsHans G. Kaper, Man Kam Kwong, and Anton ZettlHans G. Kaper, Man Kam Kwong, and Anton Zettlhttps://doi.org/10.1137/0517056PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented to characterize selfadjoint realizations of a singular Sturm–Liouville differential expression on a finite interval, where the singularities are of limit-circle type.[1] M. A. Naimark, Linear differential operators. Part I: Elementary theory of linear differential operators, Frederick Ungar Publishing Co., New York, 1967xiii+144 35:6885 M. A. Naimark, Linear differential operators. Part II: Linear differential operators in Hilbert space, With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt, Frederick Ungar Publishing Co., New York, 1968xv+352 41:7485 Google Scholar[2] N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1961xi+147 41:9015a N. I. Akhiezer and , I. M. Glazman, Theory of linear operators in Hilbert space. Vol. II, Translated from the Russian by Merlynd Nestell, Frederick Ungar Publishing Co., New York, 1963v+218 41:9015b Google Scholar[3] H. Weyl, Üher gewöhnliche Differentialgleichungen mit Singulärituten and die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann, 68 (1910), 220–269 CrossrefGoogle Scholar[4] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vol. 2, Oxford Univ. Press, Cambridge, 1962 CrossrefGoogle ScholarKeywordsSturm–Liouville differential operatorssingularities of limit-circle typeselfadjoint realizationsFriedrichs extension Previous article Next article FiguresRelatedReferencesCited ByDetails Friedrichs extensions of a class of singular Hamiltonian systemsJournal of Differential Equations, Vol. 293 | 1 Aug 2021 Cross Ref On self-adjoint boundary conditions for singular Sturm–Liouville operators bounded from belowJournal of Differential Equations, Vol. 269, No. 9 | 1 Oct 2020 Cross Ref Friedrichs extensions for singular Hamiltonian operators with intermediate deficiency indicesJournal of Mathematical Analysis and Applications, Vol. 461, No. 2 | 1 May 2018 Cross Ref On Properties of the Legendre Differential ExpressionResults in Mathematics, Vol. 42, No. 1-2 | 16 May 2013 Cross Ref The Friedrichs Extension of Singular Differential OperatorsJournal of Differential Equations, Vol. 160, No. 2 | 1 Jan 2000 Cross Ref Density, spectral theory and homoclinics for singular Sturm-Liouville systemsJournal of Computational and Applied Mathematics, Vol. 52, No. 1-3 | 1 Jul 1994 Cross Ref Singular Second-Order Operators: The Maximal and Minimal Operators, and Selfadjoint Operators in BetweenMojdeh Hajmirzaahmad and Allan M. KrallSIAM Review, Vol. 34, No. 4 | 2 August 2006AbstractPDF (1871 KB)Eigenvalue and eigenfunction computations for Sturm-Liouville problemsACM Transactions on Mathematical Software, Vol. 17, No. 4 | 1 Dec 1991 Cross Ref Volume 17, Issue 4| 1986SIAM Journal on Mathematical Analysis761-1035 History Submitted:17 August 1984Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsSturm–Liouville differential operatorssingularities of limit-circle typeselfadjoint realizationsFriedrichs extensionMSC codes34B2547E05PDF Download Article & Publication DataArticle DOI:10.1137/0517056Article page range:pp. 772-777ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

<i>Theory of Ordinary Differential Equations</i>

1956-02-01
Earl A. Coddington , Norman Levinson , T. Teichmann
The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by … The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.

Strongly singular potentials and essential self-adjointness of singular elliptic operators in C0∞(Rn⧹{0})

1972-05-01
Hubert Kalf , Johann Walter

Mathematical Methods in Quantum Mechanics

2009-04-01
Gerald Teschl
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain … Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.

The Friedrichs Extension of Singular Differential Operators

2000-01-01
Marco Marletta , Anton Zettl

Orthogonality of Jacobi polynomials with general parameters

2003-01-01
Arno B. J. Kuijlaars , Andrei Martı́nez-Finkelshtein , R. Orive
In this paper we study the orthogonality conditions satisfied by Jacobi polynomials $P_n^{(α,β)}$ when the parameters $α$ and $β$ are not necessarily $&gt;-1$. We establish orthogonality on a generic closed … In this paper we study the orthogonality conditions satisfied by Jacobi polynomials $P_n^{(α,β)}$ when the parameters $α$ and $β$ are not necessarily $&gt;-1$. We establish orthogonality on a generic closed contour on a Riemann surface. Depending on the parameters, this leads to either full orthogonality conditions on a single contour in the plane, or to multiple orthogonality conditions on a number of contours in the plane. In all cases we show that the orthogonality conditions characterize the Jacobi polynomial $P_n^{(α, β)}$ of degree $n$ up to a constant factor.

Singular boundary conditions for Sturm–Liouville operators via perturbation theory

2022-06-23
Michael D. Bush , Dale Frymark , Constanze Liaw
We show that all self-adjoint extensions of semi-bounded Sturm--Liouville operators with general limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the … We show that all self-adjoint extensions of semi-bounded Sturm--Liouville operators with general limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in\{1,2\}$. This characterization generalizes the well-known analog for semi-bounded Sturm--Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as \begin{align*} \boldsymbol{A}_\Theta=\boldsymbol{A}_0+{\bf B}\Theta{\bf B}^*, \end{align*} where $\boldsymbol{A}_0$ is a distinguished self-adjoint extension and $\Theta$ is a self-adjoint linear relation in $\mathbb{C}^d$. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to $\boldsymbol{A}_0$, i.e. it belongs to $\mathcal{H}_{-1}(\boldsymbol{A}_0)$. The construction of a boundary triple and compatible boundary pair for the symmetric operator ensure that the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations $\Theta$. As an example, self-adjoint extensions of the classical symmetric Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.
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The Sturm-Liouville Friedrichs extension

2015-06-01
Siqin Yao , Jiong Sun , Anton Zettl
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Some Applications of Operator-valued Herglotz Functions

2001-01-01
Fritz Gesztesy , N. J. Kalton , Konstantin A. Makarov +1 more
We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both … We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for classes of Herglotz operators. Moreover, we study the concrete case of Schrödinger operators on a half-line and provide two illustrations of Livšic's result [] on quasi-hermitian extensions in the special case of densely defined symmetric operators with deficiency indices (,).
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Spectral Theory and Differential Operators

2018-05-23
David E. Edmunds , Des Evans
Abstract This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential … Abstract This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential operators, and makes applications to such questions. After the exposition of the abstract theory in the first four chapters, Sobolev spaces are introduced and their main properties established. The remaining seven chapters are largely concerned with second-order elliptic differential operators and related boundary-value problems. Particular attention is paid to the spectrum of the Schrödinger operator. Its original form contains material of lasting importance that is relatively unaffected by advances in the theory since 1987, when the book was first published. The present edition differs from the old by virtue of the correction of minor errors and improvements of various proofs. In addition, it contains Notes at the ends of most chapters, intended to give the reader some idea of recent developments together with additional references that enable more detailed accounts to be accessed.

A Survey of Some Norm Inequalities

2021-01-27
Fritz Gesztesy , Roger Nichols , Jonathan Stanfill
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Functional determinants for general Sturm–Liouville problems

2004-04-05
Klaus Kirsten , Alan J. McKane
Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems … Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm-Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses the properties of generalised zeta-functions. The general form of the final results are the same for the entire range of problems considered.
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Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

2013-01-01
Jonathan Eckhardt , Fritz Gesztesy , Roger Nichols +1 more
We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ \tau f = \frac{1}{r} (- … We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ \tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).
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Weyl function and spectral properties of self-adjoint extensions

2002-09-01
Johannes F. Brasche , M. M. Malamud , Hagen Neidhardt
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On a Class of Integral Inequalities

1978-04-01
W. N. Everitt , Anton Zettl

On the singular Weyl–Titchmarsh function of perturbed spherical Schrödinger operators

2010-11-11
Aleksey Kostenko , Gerald Teschl
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Boundary value problems for elliptic partial differential operators on bounded domains

2006-11-29
Jussi Behrndt , Matthias Langer
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Boundary values of analytic operator functions with a positive imaginary part

1989-03-01
Serguei Naboko

An Addendum to Krein's Formula

1998-06-01
Fritz Gesztesy , Konstantin A. Makarov , Eduard Tsekanovskiı̆
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The extension theory of Hermitian operators and the moment problem

1995-01-01
Volodymyr Derkach , M. M. Malamud

Non-self-adjoint operators, infinite determinants, and some applications

2005-01-01
Fritz Gesztesy , Yuri Latushkin , Marius Mitrea +1 more
We study various spectral theoretic aspects of non-self-adjoint operators. Specifically, we consider a class of factorable non-self-adjoint perturbations of a given unperturbed non-self-adjoint operator and provide an in-depth study of … We study various spectral theoretic aspects of non-self-adjoint operators. Specifically, we consider a class of factorable non-self-adjoint perturbations of a given unperturbed non-self-adjoint operator and provide an in-depth study of a variant of the Birman-Schwinger principle as well as local and global Weinstein-Aronszajn formulas. Our applications include a study of suitably symmetrized (modified) perturbation determinants of Schrödinger operators in dimensions n=1,2,3 and their connection with Krein's spectral shift function in two- and three-dimensional scattering theory. Moreover, we study an appropriate multi-dimensional analog of the celebrated formula by Jost and Pais that identifies Jost functions with suitable Fredholm (perturbation) determinants and hence reduces the latter to simple Wronski determinants.

The Bessel differential equation and the Hankel transform

2006-12-06
W. N. Everitt , Hubert Kalf

Spectral Functions in Mathematics and Physics

2001-12-13
Klaus Kirsten
Spectral functions relevant in the context of quantum field theory under the influence of spherically symmetric external conditions are analysed.Examples comprise heatkernels, determinants and spectral sums needed for the analysis … Spectral functions relevant in the context of quantum field theory under the influence of spherically symmetric external conditions are analysed.Examples comprise heatkernels, determinants and spectral sums needed for the analysis of Casimir energies.First, we summarize that a convenient way of handling them is to use the associated zeta function.A way to determine all its needed properties is derived.Using the connection with the mentioned spectral functions, we provide: i.) a method for the calculation of heat-kernel coefficients of Laplace-like operators on Riemannian manifolds with smooth boundaries and ii.) an analysis of vacuum energies in the presence of spherically symmetric boundaries and external background potentials.
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Functional determinants by contour integration methods

2003-07-31
Klaus Kirsten , Alan J. McKane
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Ten Physical Applications of Spectral Zeta Functions

2012-01-01
E. Elizalde