Interface Problems for Dispersive equations

TL;DR

This paper analyzes interface problems for the linear Schrödinger equation in one-dimensional piecewise homogeneous domains, providing explicit solutions and addressing challenges posed by the dispersive nature of the equation.

Contribution

It extends previous methods to handle dispersive Schrödinger interface problems with explicit solutions for both finite and semi-infinite domains.

Findings

Explicit solutions for interface problems in Schrödinger equations

Handling dispersive effects at interfaces

Extension of previous methods to new domain configurations

Abstract

The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. The problem and the method considered here extend that of an earlier paper by Deconinck, Pelloni and Sheils (2014). The dispersive nature of the problem presents additional difficulties that are addressed here.