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Quantum Studies Mathematics and Foundations

Evolution of Aharonov–Berry superoscillations in Dirac $$\varvec{\delta }$$ δ -potential

Jussi Behrndt, Fabrizio Colombo, Peter Schlosser

Published Mar 18, 2019• 24 citations
Publisher

Abstract

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.

DOI
10.1007/s40509-019-00188-4

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Schrödinger evolution of superoscillations with δ\delta δ- and δ′\delta 'δ′-potentials
Abstract In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with δ\delta δ δ - and δ′\delta 'δ′ δ ′ -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 A1(C)A_1(\mathbb {C})A1​(C) A 1 ( 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 δ\delta δ δ - and δ′\delta 'δ′ δ ′ -potentials.
Time evolution of superoscillations for the Schrödinger equation on R∖{0}{\mathbb {R}}\setminus \{0\}R∖{0}
Abstract In quantum mechanics, superoscillations, or the more general supershifts, appear as initial conditions of the time-dependent Schrödinger equation. Already in [5], a unified approach was developed, which yields time persistence of the supershift property under certain holomorphicity and growth assumptions on the corresponding Green’s function. While that theory considers the Schrödinger equation on the whole real line R{\mathbb {R}}R R , this paper takes the natural next step and considers R∖{0}\mathbb {R}\setminus \{0\}R∖{0} R \ { 0 } , while allowing boundary conditions at x=0±x=0^\pm x=0± x = 0 ± . In particular, the singular 1x2\frac{1}{x^2}x21​ 1 x 2 -potential as well as the very important δ\delta δ δ and δ′\delta 'δ′ δ ′ distributional potentials are covered.
The path-integral technique in quantum mechanics provides an intuitive framework for comprehending particle propagation and scattering. Calculating the propagator for the Aharonov-Bohm potential fits into the range of potentials in multiply-connected spaces, with the propagator represented through a series expansion. In this paper, we analyze the Schrödinger evolution of superoscillations, showing the supershift properties of the solution to the Schrödinger equation for this potential. Our proof is based on the continuity of particular infinite order differential operators acting on spaces of entire functions.
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.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger 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 one-dimensional Schrödinger 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.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger 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 one-dimensional Schrödinger 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.