Resonances and Spectral Shift Function for a Magnetic SCHR\"Odinger Operator

We consider the 3D Schr"odinger operator $H_0$ with constant magnetic field and subject to an electric potential $v_0$ depending only on the variable along the magnetic field $x_3$. The operator $H_0$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H_0$ by smooth scalar potentials $V=O((x_1,x_2){>}^{-{\text{de}}_{\perp }}{x}_3{>}^{-{\text{de}}_{\parallel }})$, ${\text{de}}_{\perp }>2,{\text{de}}_{\parallel }>1$. We assume also that $V$ and $v_0$ have an analytic continuation, in the magnetic field direction, in a complex sector outside a compact set. We define the resonances of $H=H_0+V$ as the eigenvalues of the non-selfadjoint operator obtained from $H$ by analytic distortions of ${\mathbb{R}}_{x_3}$. We study their distribution near any fixed real eigenvalue of $H_0$, $2bq+\text{la}$ for $q\in \mathbb{N}$. In a ring centered at $2bq+\text{la}$ with radiuses $(r,2r)$, we establish an upper bound, as $r$ tends to 0, of the number of resonances. This upper bound depends on the decay of $V$ at infinity only in the directions $(x_1,x_2)$. Finally, we deduce a representation of the derivative of the spectral shift function (SSF) for the operator pair ($H_0,H$) in terms of resonances. This representation justifies the Breit-Wigner approximation and implies a local trace formula.