Counting negative eigenvalues of the Neumann Pauli Operator in bounded domains

In 1979, Aharonov and Casher provided a precise formula for the zero eigenspace dimension of the Pauli operator. Since then, researchers have explored various generalizations and variations of the theorem. Recently, we have studied a case in which the Neumann or Robin condition is imposed on the boundary of a bounded domain in the plane, leading to potential negative eigenvalues. Our goal is to accurately count these negative eigenvalues. While we provide an exact formula for the disc, we can only offer a lower bound for general domains. Furthermore, we examine some semi-classical implications of our findings. Joint work with Rupert Frank, Magnus Goffeng, Ayman Kachmar, and Mikael Persson Sundqvist.