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

TL;DR

This paper analyzes the asymptotic behavior of eigenvalue counts and Riesz means for Pauli and Dirac operators under large magnetic fields as Planck's constant approaches zero, extending previous results with new uniformity conditions.

Contribution

It provides new semi-classical asymptotic formulas for these operators with large magnetic fields, covering arbitrary directions and uniform cases, complementing prior work by Sobolev.

Findings

Asymptotic formulas for eigenvalue counts as 4040 with large magnetic fields.

Results hold for arbitrary magnetic field directions and are uniform in field strength.

Extends previous work by Sobolev with new uniformity and direction considerations.

Abstract

We study the asymptotic behavior, as Planck's constant $\hbar\to 0$, 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 $\hbar\to 0$. Two main types of results are established: in the first $\mu\hbar\le constant$ as $\hbar\to 0$, 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.