R\'esonances pr\`es de seuils d'op\'erateurs magn\'etiques de Pauli et de Dirac

TL;DR

This paper studies the distribution and accumulation of resonances near thresholds for perturbed magnetic Pauli and Dirac operators with super-exponentially decaying potentials, providing bounds and asymptotic behaviors.

Contribution

It introduces a framework for meromorphic extension of resolvents of magnetic Pauli and Dirac operators and analyzes resonance distribution near thresholds.

Findings

Resonances are characterized as poles of meromorphic resolvent extensions.

Upper bounds on the number of resonances near thresholds are established.

Asymptotic expansions describe resonance accumulation near thresholds.

Abstract

We consider the perturbations $H := H_{0} + V$ and $D := D_{0} + V$ of the free 3D Hamiltonians $H_{0}$ of Pauli and $D_{0}$ of Dirac with non-constant magnetic field, and $V$ is a electric potential which decays super-exponentially with respect to the variable along the magnetic field. We show that in appropriate Banach spaces, the resolvents of $H$ and $D$ defined on the upper half-plane admit meromorphic extensions. We define the resonances of $H$ and $D$ as the poles of these meromorphic extensions. We study the distribution of resonances of $H$ close to the origin 0 and that of $D$ close to $\pm m$, where $m$ is the mass of a particle. In both cases, we first obtain an upper bound of the number of resonances in small domains in a vicinity of 0 and $\pm m$. Moreover, under additional assumptions, we establish asymptotic expansions of the number of resonances which imply their accumulation near the thresholds 0 and $\pm m$. In particular, for a perturbation $V$ of definite sign, we obtain information on the distribution of eigenvalues of $H$ and $D$ near 0 and $\pm m$ respectively.