Counting function of magnetic eigenvalues for non-definite sign perturbations

TL;DR

This paper analyzes the eigenvalues of a 3D Pauli operator with a non-constant magnetic field perturbed by a non-definite sign electric potential, revealing their concentration and asymptotic behavior near zero energy.

Contribution

It provides new results on the distribution and asymptotics of eigenvalues for Pauli operators with non-definite sign perturbations near zero energy.

Findings

Eigenvalues are concentrated in the negative semi-axis near zero.

Resonances are only eigenvalues close to the ground energy zero.

New asymptotic expansions and bounds on eigenvalue count near zero.

Abstract

We consider the perturbed operator $H(b,V) := H(b,0) + V$, where $H(b,0)$ is the $3$d Hamiltonian of Pauli with non-constant magnetic field, and $V$ is \textit{a non-definite sign electric potential} decaying exponentially with respect to the variable along the magnetic field. We prove that the only resonances of $H(b,V)$ near the low ground energy zero of $H(b,0)$ are its eigenvalues and are concentrated in the semi axis $(-\infty,0)$. Further, we establish new asymptotic expansions, upper and lower bounds on their number near zero.