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We consider the Pauli operator $\mathbf{H}\left(\mu \right):={\left({\sum }_{j=1}^m{\sigma }_j\left(-i\frac{\partial }{\partial x_j}-\mu A_j\right)\right)}^2+V$ selfadjoint in $L^2({ℝ}^m;{ℂ}^2)$, $m=2,3$. Here ${\sigma }_j$, $j=1,...,m$, are the Pauli matrices, $A:=(A_1,...,A_m)$ is the magnetic potential, $\mu \>0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m=3$, its direction is constant. We investigate the asymptotic behaviour as $\mu \to \infty$ of the number of the eigenvalues of $\mathbf{H}\left(\mu \right)$ smaller than...

We consider the 3D Schrödinger operator $H=H_0+V$ where $H_0={(-i\nabla -A)}^2-b$, $A$ is a magnetic potential generating a constant magneticfield of strength $b\>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a meromorphic extension from the upper half plane to an appropriate Riemann surface $ℳ$, and define the resonances of $H$ as the poles of this meromorphic extension. We study their distribution near any fixed...

We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic...