Counting negative eigenvalues with index theory

In this talk we will consider the Pauli Hamiltonian, a Schrödinger operator describing a spin particle moving in a magnetic field in two dimensions. The problem we will discuss is that of describing how many negative eigenvalues there are for its Neumann realization on a compact domain. We provide a sharp lower bound for the number of negative eigenvalues by means of a connection to Atiyah-Patodi-Singer’s index theorem for Dirac operators arising from an elementary integration by parts. We conclude an asymptotic formula for the counting function of the semiclassical Landau-Neumann Hamiltonian below the first Landau level as hbar goes to zero.

Joint work with Søren Fournais, Rupert Frank, Ayman Kachmar and Mikael Persson-Sundqvist.