Spectral estimates for Dirichlet Laplacians on perturbed twisted tubes

TL;DR

This paper studies the spectral properties of Dirichlet Laplacians in twisted tubes with non-circular cross sections, deriving bounds on the discrete spectrum influenced by the tube's geometry.

Contribution

It provides a Lieb-Thirring-type estimate for spectral moments in twisted tubes and illustrates how the bounds depend on the cross section shape.

Findings

Derived spectral bounds depend on the tube's cross section shape.

Established a Lieb-Thirring-type inequality for the discrete spectrum.

Presented examples demonstrating the influence of geometric perturbations.

Abstract

We investigate Dirichlet Laplacian in a straight twisted tube of a non-circular cross section, in particular, its discrete spectrum coming from a local slowdown of the twist. We prove a Lieb-Thirring-type estimate for the spectral moments and present two examples illustrating how the bound depends on the tube cross section.