2012/01/31 by Diomba Sambou, Sambou, Diomba
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1201.6552
25 pages
arxiv created 2012/11/11 · arxiv updated 2012/11/13
We consider the perturbations H := H0 + V and D := D0 + V of the free 3D Hamiltonians H0 of Pauli and D0 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 ± 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 ± m. Moreover, under additional assumptions, we establish asymptotic expansions of the number of resonances which imply their accumulation near the thresholds 0 and ± 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 ± m respectively.