Ground Energy of the Magnetic Laplacian in Polyhedral Bodies

TL;DR

This paper analyzes the asymptotic behavior of the first eigenvalues of magnetic Laplacians in polyhedral domains with large magnetic fields, introducing new methods for bounds and eigenfunction properties.

Contribution

It provides a detailed study of model problems and introduces a novel construction of quasimodes for estimating eigenvalues in polyhedral domains.

Findings

Established estimates for asymptotic remainders.

Derived lower bounds using IMS partitioning.

Developed a new construction of quasimodes for upper bounds.

Abstract

The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields and Neumann realization in polyhedral domains is characterized by a hierarchy of model problems. We investigate properties of the model problems (continuity, semi-continuity, existence of generalized eigenfunctions). We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with the help of a classical IMS partition based on adequate coverings of the polyhedral domain, whereas upper bounds are established by a novel construction of quasimodes, qualified as sitting or sliding according to spectral properties of local model problems.