Spectral estimates for Dirichlet Laplacians and Schroedinger operators on geometrically nontrivial cusps

TL;DR

This paper derives eigenvalue estimates for Dirichlet Laplacians and Schrödinger operators in complex, curved, or twisted cusps, revealing how geometry influences spectral bounds and can outperform classical semiclassical estimates.

Contribution

It introduces new eigenvalue bounds that incorporate geometric features of nontrivial cusps, extending spectral theory to more complex geometries.

Findings

Eigenvalue bounds depend on cusp geometry.

Bounds can surpass semiclassical estimates in certain energy regions.

Cusps' geometric properties significantly influence spectral estimates.

Abstract

The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schr\"odinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.