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

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.