Semiclassical bounds in magnetic bottles

TL;DR

This paper derives spectral bounds for magnetic systems, including regions with boundaries and magnetic field variations, providing new eigenvalue estimates for magnetic Laplacians in three dimensions.

Contribution

It introduces new Berezin-Li-Yau and Lieb-Thirring-type bounds for magnetic Laplacians in both two and three dimensions, expanding spectral estimate techniques.

Findings

Established two-dimensional spectral bounds with magnetic fields.

Derived three-dimensional eigenvalue moment estimates.

Extended spectral estimate methods to magnetic systems with boundaries.

Abstract

The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in $\mathbb{R}^3$ confined by a local change of the magnetic field. We establish two-dimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.