Lieb-Thirring inequalities for geometrically induced bound states

TL;DR

This paper establishes optimal Lieb-Thirring inequalities for Schrödinger operators in wave guides with local perturbations, providing insights into bound states in geometrically modified structures.

Contribution

It introduces new Lieb-Thirring inequalities tailored for wave guides with boundary deformations, enhancing understanding of eigenvalue bounds in such geometries.

Findings

Optimal inequalities in weak-coupling regimes

Applications to straight strips and circular tubes

Analysis of boundary condition effects

Abstract

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schr\"odinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.