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

TL;DR

This paper investigates the distribution of resonances and the spectral shift function for a 3D magnetic Schrödinger operator with electric potential, establishing bounds and representations related to spectral perturbations.

Contribution

It introduces a detailed analysis of resonances near embedded eigenvalues for magnetic Schrödinger operators with complex-analytic potentials, including bounds and spectral shift function representations.

Findings

Upper bounds on resonance counts near fixed eigenvalues

Representation of the spectral shift function derivative in terms of resonances

Justification of the Breit-Wigner approximation and local trace formula

Abstract

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.