Energy decay for a system of Schr{\"o}dinger equations in a wave guide

TL;DR

This paper proves exponential energy decay in a coupled Schr{"o}dinger system within a wave guide, utilizing spectral analysis to establish a spectral gap and basis properties of eigenfunctions.

Contribution

It introduces a novel spectral analysis approach for coupled Schr{"o}dinger equations with boundary damping, demonstrating exponential decay in wave guides.

Findings

Established exponential decay of energy in the system.

Proved the existence of a spectral gap for the coupled operator.

Showed that generalized eigenfunctions form a Riesz basis.

Abstract

We prove exponential decay for a system of two Schr{\"o}dinger equations in a wave guide, with coupling and damping at the boundary. This relies on the spectral analysis of the corresponding coupled Schr{\"o}dinger operator on the one-dimensional cross section. We show in particular that we have a spectral gap and that the corresponding generalized eigenfunctions form a Riesz basis.