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Green's function for the Schrodinger equation with a generalized point interaction and stability of superoscillations

Yakir Aharonov, Jussi Behrndt, Fabrizio Colombo, Peter Schlosser

Published Jan 1, 2021• 22 citations
Publisher

Abstract

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.

DOI
10.1016/j.jde.2020.12.029

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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.
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.
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.
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.
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrodinger's equation. This paper, in contrast, investigates the integral form of Schrodinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrodinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrodinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schrodinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrodinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrodinger's integral equation is viable tool for studying singular interactions in quantum mechanics.
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.
Weak measurements and the theory of weak values have a very deep meaning in quantum mechanics, and new phenomena associated with them has recently been observed experimentally. This theory has also directly led to the notion of superoscillating sequences of functions. In this paper we consider Cauchy problems with superoscillatory initial conditions (in particular, the Cauchy problem for the Schrödinger equation and some of its variations), and we give conditions under which the superoscillations persist in time. Our work is based on results from the theory of formal solutions to Cauchy problems and on the study of the specific growth of superoscillatory sequences, when regarded as functions of a complex variable. There are two main aims of this paper: one is to explain the mathematical tools that are necessary to study superoscillations, also repeating a few results that we have already proved in other papers in order to clarify the strategy. The second aim is to show that our technique applies to a large class of problems, showing under which conditions the superoscillatory phenomenon persists. Finally, we point out that our strategy can be applied also to non-constant coefficients differential equations as the quantum harmonic oscillator.