Direct and inverse problems for the Schr\"{o}dinger operator in a three-dimensional planar waveguide

TL;DR

This paper investigates the spectral properties of the Schrödinger operator in a 3D waveguide, establishing resonance-free regions, resolvent bounds, and stability results for inverse source problems using limited frequency data.

Contribution

It provides new results on the meromorphic continuation, resonance-free regions, and stability estimates for inverse problems in a 3D waveguide setting.

Findings

Existence of a resonance-free region for the resolvent.

Unique solvability of the direct source problem.

Increasing stability for the inverse source problem with high-frequency data.

Abstract

In this paper, we study the meromorphic continuation of the resolvent for the Schr\"{o}dinger operator in a three-dimensional planar waveguide. We prove the existence of a resonance-free region and an upper bound for the resolvent. As an application, the direct source problem is shown to have a unique solution under some appropriate assumptions. Moreover, an increasing stability is achieved for the inverse source problem of the Schr\"{o}dinger operator in the waveguide by using limited aperture Dirichlet data only at multiple frequencies. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases.