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

Yakir Aharonov, Jussi Behrndt, Fabrizio Colombo, Peter Schlosser

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)153-190
Number of pages38
JournalJournal of Differential Equations
Volume277
DOIs
StatePublished - 15 Mar 2021
Externally publishedYes

Keywords

  • Green's function
  • Point interaction
  • Schrödinger equation
  • Superoscillating function

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