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 $V=O((x_1,x_2)>^{-\de_\perp}x_3>^{-\de_\parallel})$, $\de_\perp>2, \de_\parallel>1$. 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 $\R_{x_3}$. We study their distribution near any fixed real eigenvalue of $H_0$, $2bq+\la$ for $q\in\N$. In a ring centered at $2bq+\la$ 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.