Semi-classical asymptotics for the counting functions and Riesz means of Pauli and Dirac operators with large magnetic fields

Abstract

We study the asymptotic behavior, as Planck's constant $\hslash \to 0$, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field $\mu \mathbf{B}(x)$ and an electric field. The magnetic field strength $\mu$ is allowed to tend to infinity as $\hslash \to 0$. Two main types of results are established: in the first $\mu \hslash \le constant$ as $\hslash \to 0$, with magnetic fields of arbitrary direction; the second results are uniform with respect to $\mu \ge 0$ but the magnetic fields have constant direction. The results on the Pauli operator complement recent work of Sobolev.