Discrete Schrödinger operators with decaying and oscillating potentials

Abstract

The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V left-parenthesis n right-parenthesis equals lamda n Superscript negative alpha Baseline cosine left-parenthesis pi omega n Superscript beta Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> <mml:mi>cos</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>π<!-- π --></mml:mi> <mml:mi>ω<!-- ω --></mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mi>β<!-- β --></mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">V(n)=\lambda n^{-\alpha }\cos (\pi \omega n^\beta )</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="1 greater-than beta greater-than 2 alpha"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1&gt;\beta &gt;2\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.

Locations

• St Petersburg Mathematical Journal - View
• arXiv (Cornell University) - View - PDF
• CaltechAUTHORS (California Institute of Technology) - View - PDF