The aim of this article is to introduce the H^{\infty} -functional calculus for unbounded bisectorial operators in a Clifford module over the algebra \mathbb{R}_{n} . This work is based on …
The aim of this article is to introduce the H^{\infty} -functional calculus for unbounded bisectorial operators in a Clifford module over the algebra \mathbb{R}_{n} . This work is based on the universality property of the S -functional calculus, which shows its applicability to fully Clifford operators. While recent studies have focused on bounded operators or unbounded paravector operators, we now investigate unbounded fully Clifford operators and define polynomially growing functions of them. We first generate the \omega -functional calculus for functions that exhibit an appropriate decay at zero and at infinity. We then extend to functions with a finite value at zero and at infinity. Finally, using a subsequent regularization procedure, we can define the H^{\infty} - functional calculus for the class of regularizable functions, which in particular include functions with polynomial growth at infinity and, if T is injective, also functions with polynomial growth at zero.
Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides …
Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum.
In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier's law for heat propagation and Fick's first law, that relates the diffusive flux to the gradient …
In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier's law for heat propagation and Fick's first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider $n\geq 3$ orthogonal unit vectors $e_1,\dots,e_n\in\mathbb{R}^n$, and let $\Omega\subseteq\mathbb{R}^n$ be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator $T=\sum_{i=1}^ne_ia_i(x)\frac{\partial}{\partial x_i}$ with nonconstant positive coefficients $a_i:\overline{\Omega}\to(0,\infty)$. Under certain regularity and growth conditions on the $a_i$, we identify bisectorial or strip-type regions that belong to the $S$-resolvent set of $T$. Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the $S$-spectrum, designed to study the operators acting in Clifford modules $V$ over the Clifford algebra $\mathbb{R}_n$, with vector operators being a specific crucial subclass. The spectral properties related to the $S$-spectrum of $T$ are linked to the inversion of the operator $Q_s(T):=T^2-2s_0T+|s|^2$, where $s\in\mathbb{R}^{n+1}$ is a paravector, i.e., it is of the form $s=s_0+s_1e_1+\dots+s_ne_n$. This spectral problem is substantially different from the complex one, since it allows to associate general boundary conditions to $Q_s(T)$, i.e., to the squared operator $T^2$.
The aim of this paper is to introduce the H^{\infty} -functional calculus for harmonic functions over the quaternions. More precisely, we give meanring to Df(T) for unbounded sectorial operators T …
The aim of this paper is to introduce the H^{\infty} -functional calculus for harmonic functions over the quaternions. More precisely, we give meanring to Df(T) for unbounded sectorial operators T and polynomially growing functions of the form Df , where f is a slice hyperholomorphic function and D=\partial_{q_0}+e_{1}\partial_{q_1}+e_{2}\partial_{q_2}+e_{3}\partial_{q_3} is the Cauchy–Fueter operator. The harmonic functional calculus can be viewed as a modification of the well-known S -functional calculus f(T) , with a different resolvent operator. The harmonic H^{\infty} -functional calculus is defined in two steps. First, for functions with a certain decay property, one can make sense of the bounded operator Df(T) directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators Df(T) . The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the S -spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter–Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hypercomplex analysis.
This paper is inspired by a class of infinite order differential operators arising in the time evolution of superoscillations. Recently, infinite order differential operators have been considered and characterized on …
This paper is inspired by a class of infinite order differential operators arising in the time evolution of superoscillations. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e., functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra. We introduce the concept of proximate order and establish some fundamental properties of entire hyperholomorphic functions that are crucial for this characterization.
We consider the self-adjoint Schrodinger operator in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $\delta$-potential supported on a hyperplane $\Sigma\subseteq\mathbb{R}^d$ of strength $\alpha=\alpha_0+\alpha_1$, where $\alpha_0\in\mathbb{R}$ is a constant and $\alpha_1\in L^p(\Sigma)$ is …
We consider the self-adjoint Schrodinger operator in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $\delta$-potential supported on a hyperplane $\Sigma\subseteq\mathbb{R}^d$ of strength $\alpha=\alpha_0+\alpha_1$, where $\alpha_0\in\mathbb{R}$ is a constant and $\alpha_1\in L^p(\Sigma)$ is a nonnegative function. As the main result, we prove that the lowest spectral point of this operator is not smaller than that of the same operator with potential strength $\alpha_0+\alpha_1^*$, where $\alpha_1^*$ is the symmetric decreasing rearrangement of $\alpha_1$. The proof relies on the Birman-Schwinger principle and the reduction to an analogue of the Polya-Szegő inequality for the relativistic kinetic energy in $\mathbb{R}^{d-1}$.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrodinger equation is an important and challenging problem in quantum …
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrodinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper we give a unified approach to determine the supershift property for the solution of the time dependent Schrodinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique we are able to treat various potentials.
In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary …
In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
In this paper we study the time dependent Schrodinger equation with all possible self-adjoint singular interactions located at the origin, which include the $\delta$ and $\delta'$-potentials as well as boundary …
In this paper we study the time dependent Schrodinger equation with all possible self-adjoint singular interactions located at the origin, which include the $\delta$ and $\delta'$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases.
We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrodinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
Abstract In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> - and $$\delta …
Abstract In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> - and $$\delta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>δ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> -potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converge in the topology of the entire function space $$A_1(\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> - and $$\delta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>δ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> -potentials.
In this paper we study the time persistence of superoscillations as the initial data of the time dependent Schr\odinger equation with $\delta$- and $\delta'$-potentials. It is shown that the sequence …
In this paper we study the time persistence of superoscillations as the initial data of the time dependent Schr\odinger equation with $\delta$- and $\delta'$-potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converges in the topology of the entire function space $A_1(\mathbb{C})$. Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under $\delta$- and $\delta'$-potentials.
The main goal of this note is to study the time evolution of superoscillations under the 1D-Schrödinger equation with attractive or repulsive Dirac $$\delta $$ -potential located at the origin …
The main goal of this note is to study the time evolution of superoscillations under the 1D-Schrödinger equation with attractive or repulsive Dirac $$\delta $$ -potential located at the origin of the real line. Such potentials are of particular interest since they simulate short range interactions and the corresponding quantum system is an explicitly solvable model. Moreover, we give the large time asymptotics of this solution, which turns out to be different for the repulsive and the attractive model. The method that we use to study the time evolution of superoscillations is based on the continuity of the time evolution operator acting in a space of exponentially bounded entire functions.
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations.Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak …
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations.Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics.From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum.Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces.In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
The time-dependent Schrödinger equation with a time-dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation …
The time-dependent Schrödinger equation with a time-dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wavefunction at the origin, one may derive the wavefunction everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the potential lead to the conservation of the normalization of the probability density.
It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible - a phenomenon called superoscillation. Here, we consider the case of …
It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible - a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.
It is commonly assumed that a signal bandlimited to mu/2 Hz cannot oscillate at frequencies higher than mu Hz. In fact, however, for any fixed bandwidth, there exist finite energy …
It is commonly assumed that a signal bandlimited to mu/2 Hz cannot oscillate at frequencies higher than mu Hz. In fact, however, for any fixed bandwidth, there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients, called superoscillations, can only occur in signals that possess amplitudes of widely different scales. This paper investigates the required dynamical range and energy (squared L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 </sup> norm) as a function of the superoscillation's frequency, number, and maximum derivative. It briefly discusses some of the implications of superoscillating signals, in reference to information theory and time-frequency analysis, for example. It also shows, among other things, that the required energy grows exponentially with the number of superoscillations, and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations' period
In this paper, we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and …
In this paper, we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schrödinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.
Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum mechanics, …
Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum mechanics, in optics, and in signal processing. An important issue is the study of the evolution of superoscillations using the Schrödinger equation when the initial datum is a weak value. Some superoscillatory functions are not square integrable, but they are real analytic functions that can be extended to entire holomorphic functions. This fact leads to the study of the continuity of a class of convolution operators acting on suitable spaces of entire functions with growth conditions. In this paper, we study the evolution of a superoscillatory initial datum in a uniform magnetic field. Moreover, we collect some results on convolution operators that appear in the theory of superoscillatory functions using a direct approach that allows the convolution operators to have non-constant coefficients of polynomial type.
We analyze a family of singular Schrödinger operators with local singular interactions supported by a hypersurface Σ ⊂ ℝn, n ≥ 2, being the boundary of a Lipschitz domain, bounded …
We analyze a family of singular Schrödinger operators with local singular interactions supported by a hypersurface Σ ⊂ ℝn, n ≥ 2, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of Σ the interaction is characterized by four real parameters, the earlier studied case of δ- and δ′-interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.
In this paper we discuss the approximation of hyperfunctions in one variable using sequences of low frequency hyperfunctions. In particular, we show that this superoscillation property holds for compactly supported …
In this paper we discuss the approximation of hyperfunctions in one variable using sequences of low frequency hyperfunctions. In particular, we show that this superoscillation property holds for compactly supported hyperfunctions. Finally, we show that the approximation is also possible for tempered hyperfunctions.
Abstract In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> - and $$\delta …
Abstract In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> - and $$\delta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>δ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> -potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converge in the topology of the entire function space $$A_1(\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> - and $$\delta '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>δ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> -potentials.
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.
We find a new representation of all boundary conditions corresponding to the so-called point interactions. We show that there is one-to-one correspondence between one-dimensional point interactions and boundary conditions of …
We find a new representation of all boundary conditions corresponding to the so-called point interactions. We show that there is one-to-one correspondence between one-dimensional point interactions and boundary conditions of the form
This text is a part of an unfinished project which deals with the generalized point interaction (GPI) in one dimension. We employ two natural parametrizations, which are known but have …
This text is a part of an unfinished project which deals with the generalized point interaction (GPI) in one dimension. We employ two natural parametrizations, which are known but have not attracted much attention, to express the resolvent of the GPI Hamiltonian as well as its spectral and scattering properties. It is also shown that the GPI yields one of the simplest models in which a non-trivial Berry phase is exhibited. Furthermore, the generalized Kronig-Penney model corresponding to the GPI is discussed. We show that there are three different types of the high-energy behaviour for the corresponding band spectrum.
The general four parameter point interaction in one dimensional quantum mechanics is regulated. It allows the exact solution, but not the perturbative one. We conjecture that this is due to …
The general four parameter point interaction in one dimensional quantum mechanics is regulated. It allows the exact solution, but not the perturbative one. We conjecture that this is due to the interaction not being asymptotically free. We then propose a different breakup of unperturbed theory and interaction, which now is asymptotically free but leads to the same physics. The corresponding regulated potential can be solved both exactly and perturbatively, in agreement with the conjecture.
In dimension greater than or equal to three, we investigate the spectrum of a Schrödinger operator with a δ-interaction supported on a cone whose cross section is the sphere of …
In dimension greater than or equal to three, we investigate the spectrum of a Schrödinger operator with a δ-interaction supported on a cone whose cross section is the sphere of codimension two. After decomposing into fibers, we prove that there is discrete spectrum only in dimension three and that it is generated by the axisymmetric fiber. We get that these eigenvalues are nondecreasing functions of the aperture of the cone and we exhibit the precise logarithmic accumulation of the discrete spectrum below the threshold of the essential spectrum.
We construct a four-parameter point-interaction for a non-relativistic particle moving on a line as the limit of a short range interaction with range tending toward zero. For particular choices of …
We construct a four-parameter point-interaction for a non-relativistic particle moving on a line as the limit of a short range interaction with range tending toward zero. For particular choices of the parameters, we can obtain a delta-interaction or the so-called delta'-interaction. The Hamiltonian corresponding to the four-parameter point-interaction is shown to correspond to the four-parameter self-adjoint Hamiltonian of the free particle moving on the line with the origin excluded.
In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary …
In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
We prove sharp Lieb-Thirring inequalities for Schrodinger operators with potentials supported on a hyperplane and we show how these estimates are related to Lieb-Thirring inequalities for relativistic Schrodinger operators.
We prove sharp Lieb-Thirring inequalities for Schrodinger operators with potentials supported on a hyperplane and we show how these estimates are related to Lieb-Thirring inequalities for relativistic Schrodinger operators.
We investigate Schr\"odinger operators with \delta- and \delta'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial …
We investigate Schr\"odinger operators with \delta- and \delta'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schr\"odinger operators with \delta- and \delta'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schr\"odinger operators and, in particular, it allows to transform known results for Schr\"odinger operators with \delta-interactions to Schr\"odinger operators with \delta'-interactions.
The main goal of this note is to study the time evolution of superoscillations under the 1D-Schrödinger equation with attractive or repulsive Dirac $$\delta $$ -potential located at the origin …
The main goal of this note is to study the time evolution of superoscillations under the 1D-Schrödinger equation with attractive or repulsive Dirac $$\delta $$ -potential located at the origin of the real line. Such potentials are of particular interest since they simulate short range interactions and the corresponding quantum system is an explicitly solvable model. Moreover, we give the large time asymptotics of this solution, which turns out to be different for the repulsive and the attractive model. The method that we use to study the time evolution of superoscillations is based on the continuity of the time evolution operator acting in a space of exponentially bounded entire functions.
We introduce a novel approach for defining a $\delta'$-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm-Liouville differential expression with measure coefficients. …
We introduce a novel approach for defining a $\delta'$-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm-Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with $\delta'$-interactions concentrated on sets of complicated structures.
Waves in a two-dimensional domain with Robin (mixed) boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena. If the variation includes a 'D point' where the boundary condition …
Waves in a two-dimensional domain with Robin (mixed) boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena. If the variation includes a 'D point' where the boundary condition is Dirichlet (vanishing wavefunction), a variety of arguments indicate that the system is singular. For a circle billiard, the boundary condition fails to determine a discrete set of levels, so the spectrum is continuous. For a diffraction grating defined by periodically varying boundary conditions on the edge of a half-plane, the phase of a diffracted beam amplitude remains undetermined. In both cases, the wavefunction on the boundary has a singularity at a D point, described by the polylogarithm function.
We re-examine the status of the weak value of a quantum mechanical observable as an objective physical concept, addressing its physical interpretation and general domain of applicability. We show that …
We re-examine the status of the weak value of a quantum mechanical observable as an objective physical concept, addressing its physical interpretation and general domain of applicability. We show that the weak value can be regarded as a definite mechanical effect on a measuring probe specifically designed to minimize the back reaction on the measured system. We then present an interesting framework for general measurement conditions (where the back reaction on the system may not be negligible) in which the measurement outcomes can still be interpreted as quantum averages of weak values. We show that in the classical limit, there is a direct correspondence between quantum averages of weak values and posterior expectation values of classical dynamical properties according to the classical inference framework.
The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-Δ-αδ(x-Γ)$ with a singular …
The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-Δ-αδ(x-Γ)$ with a singular attractive interaction supported by a graph-like set in $\mathbb{R}^ν,\: ν=2,3$. We will explain how such singular Schrödinger operators can be properly defined for different codimensions of $Γ$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $Γ$ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leaky-graph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge.
Quantum mechanics allows one to independently select both the initial and final states of a single system. Such pre- and postselection reveals novel effects that challenge our ideas about what …
Quantum mechanics allows one to independently select both the initial and final states of a single system. Such pre- and postselection reveals novel effects that challenge our ideas about what time is and how it flows.