Counting Negative Eigenvalues for the Magnetic Pauli Operator

Søren Fournais; Rupert L. Frank; Magnus Goffeng; Ayman Kachmar; Mikael Sundqvist

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arXiv · 2307.16079

Counting Negative Eigenvalues for the Magnetic Pauli Operator

Søren Fournais·Rupert L. Frank·Magnus Goffeng·Ayman Kachmar·Mikael Sundqvist

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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.

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BibTeX RIS

Søren Fournais, Rupert L. Frank, Magnus Goffeng, Ayman Kachmar, Mikael Sundqvist. 2023-07-29. Counting Negative Eigenvalues for the Magnetic Pauli Operator. https://arxiv.org/abs/2307.16079

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