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

Yakir Aharonov,Jussi Behrndt,Fabrizio Colombo,Peter Schlosser

doi:10.1016/j.jde.2020.12.029

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.

Highlights

An important problem we study in this paper is the time dependent Schrödinger equation with holomorphic initial datum F subject to a general self-adjoint singular interaction supported at the origin, that is, we consider
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. They appear in quantum mechanics as results of weak measurements and, in particular, their time evolution under the Schrödinger equation is of crucial importance, see [1,10,12,31]
Aweak measurement of a quantum observable represented by the self-adjoint operator A, involving a pre-selected state ψ0 and a post-selected state ψ1, leads to the weak value

Summary

Special cases of generalized point interactions and their Green’s functions

We consider some particular generalized point interactions and derive the explicit form of the Green’s function in these situations. In particular, that and cos(φ) = √ c and sin(φ) = √ 1 , Plugging these values in (2.19a) gives (−xy), and since we are in Case II the coefficients of the Green’s function are ω− = 0, ω+ = cot(φ) = c, μ(−x,y) = 0, μ(+x,y). We consider the free Schrödinger equation on the two half lines R \ {0} with Dirichlet boundary conditions (t, 0+) = (t, 0−) = 0, t > 0, x ∈ R \ {0}, t > 0. We consider the free Schrödinger equation on the two half lines R \ {0} with Robin boundary conditions. −b iπt in (3.5), which lead to the Green’s function (3.3)

Solution of the Schrödinger equation with a generalized point interaction

Superoscillatory initial data and plane wave asymptotics