1201.6552 (Diomba Sambou)

Résonances près de seuils d'opérateurs magnétiques de Pauli et
de Dirac    [PDF]

Diomba Sambou

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.

View original: http://arxiv.org/abs/1201.6552