A lower bound to the spectral threshold in curved quantum layers

TL;DR

This paper establishes a lower bound for the spectral threshold of the Dirichlet Laplacian in curved quantum layers, linking geometric properties of the surface to spectral properties of the Laplacian, with optimality in non-negatively curved cases.

Contribution

It introduces a geometric lower bound for the spectral threshold in curved layers, connecting surface curvature to spectral analysis, and proves its optimality under certain curvature conditions.

Findings

Lower bound depends on surface curvature and layer radius.

Bound is optimal for non-negatively curved surfaces.

Provides a spectral estimate relevant for quantum waveguides.

Abstract

We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show that it is optimal if the reference surface is non-negatively curved.