Counting eigenvalues below the lowest Landau level

TL;DR

This paper investigates how boundary conditions affect the number of eigenvalues below the lowest Landau level for magnetic Laplacians on bounded planar domains, using models like the strip and annulus.

Contribution

It quantifies the impact of mixed boundary conditions on eigenvalue counts in specific geometric models, advancing understanding of spectral properties under boundary variations.

Findings

Fewer eigenvalues below the Landau level with mixed boundary conditions.

Separation of variables reduces the problem to counting eigenvalues of fiber operators.

Quantitative analysis provided for strip and annulus models.

Abstract

For the magnetic Laplacian on a bounded planar domain, imposing Neumann boundary conditions produces eigenvalues below the lowest Landau level. If the domain has two boundary components and one imposes a Neumann condition on one component and a Dirichlet condition on the other, one gets fewer such eigenvalues than when imposing Neumann boundary conditions on the two components. We quantify this observation for two models: the strip and the annulus. In both models one can separate variables and deal with a family of fiber operators, thereby reducing the problem to counting band functions, the eigenvalues of the fiber operators.