Resonances near Thresholds in slightly Twisted Waveguides

TL;DR

This paper investigates the spectral effects of small twisting perturbations in three-dimensional waveguides, revealing the existence and asymptotic behavior of resonances near the spectrum's threshold.

Contribution

It introduces a meromorphic extension of the resolvent and characterizes the unique resonance near the spectrum's bottom caused by slight twisting.

Findings

Existence of exactly one resonance near the spectrum's threshold

Asymptotic behavior of the resonance as twisting diminishes

Bound on the number of resonances related to eigenvalue multiplicity

Abstract

We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator, and show the existence of exactly one resonance near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to the upper eigenvalues of the transversal problem, showing that the number of resonances is bounded by the multiplicity of the eigenvalue and obtaining the corresponding asymptotic behavior