Resolvent expansions and continuity of the scattering matrix at embedded thresholds: the case of quantum waveguides

TL;DR

This paper develops a resolvent expansion formula near embedded thresholds and applies it to quantum waveguides to show eigenvalue accumulation does not occur and the scattering matrix remains continuous at all thresholds.

Contribution

Introduces an inversion formula for resolvent expansions and applies it to quantum waveguides to analyze spectral properties at thresholds.

Findings

Eigenvalues do not accumulate at thresholds in the studied waveguides.

The scattering matrix is continuous at all thresholds.

The inversion formula aids in spectral analysis near embedded thresholds.

Abstract

We present an inversion formula which can be used to obtain resolvent expansions near embedded thresholds. As an application, we prove for a class of quantum waveguides the absence of accumulation of eigenvalues and the continuity of the scattering matrix at all thresholds.