Green's Function for the Schr\"odinger Equation with a Generalized Point Interaction and Stability of Superoscillations

TL;DR

This paper derives the Green's function for the Schrödinger equation with generalized point interactions, including delta potentials, and analyzes the stability and oscillations of superoscillatory solutions in quantum mechanics.

Contribution

It provides an explicit Green's function for all self-adjoint singular interactions at the origin and studies superoscillation stability in this context.

Findings

Explicit Green's function for generalized point interactions

Superoscillatory functions exhibit stability under these interactions

Analysis of oscillatory behavior of solutions with superoscillatory initial data

Abstract

In this paper we study the time dependent Schr\"odinger 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 Schr\"odinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.