Counting Negative Eigenvalues for the Magnetic Pauli Operator

TL;DR

This paper establishes a sharp lower bound on the negative eigenvalues of the magnetic Pauli operator in 2D domains, linking boundary analysis with index theory and conservation laws, and applies it to magnetic Laplacian eigenvalues.

Contribution

It introduces a novel lower bound for negative eigenvalues of the magnetic Pauli operator using boundary Dirac operators and two distinct analytical approaches.

Findings

Derived a sharp lower bound on negative eigenvalues.

Connected boundary Dirac operator analysis with eigenvalue estimates.

Applied results to magnetic Neumann Laplacian in semi-classical limit.

Abstract

We study the Pauli operator in a two-dimensional, connected domain with Neumann or Robin boundary condition. We prove a sharp lower bound on the number of negative eigenvalues reminiscent of the Aharonov-Casher formula. We apply this lower bound to obtain a new formula on the number of eigenvalues of the magnetic Neumann Laplacian in the semi-classical limit. Our approach relies on reduction to a boundary Dirac operator. We analyze this boundary operator in two different ways. The first approach uses Atiyah-Patodi-Singer index theory. The second approach relies on a conservation law for the Benjamin-Ono equation.