Counting negative eigenvalues for the magnetic Pauli operator

We study the Pauli operator in a 2-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 semiclassical 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 (APS) index theory. The second approach relies on a conservation law for the Benjamin–Ono equation.