Counting Negative Eigenvalues for the Magnetic Pauli Operator

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. This is a joint work with S. Fournais, R. L.
Frank, M. Goffeng, M. Sundqvist.