Elsevier

Journal of Differential Equations

Volume 277, 15 March 2021, Pages 153-190
Journal of Differential Equations

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

https://doi.org/10.1016/j.jde.2020.12.029Get rights and content
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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.

MSC

81Q05
35A08
32A10

Keywords

Green's function
Schrödinger equation
Point interaction
Superoscillating function

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