Spectral asymptotics for magnetic Schrödinger operator with slowly varying potential

Abstract

Consider the Schrödinger operator with constant magnetic field and smooth potential $V$ : $H(\epsilon)=H+V(\epsilon x,\epsilon y),\,\, H=D_x^2+(D_y+\mu x)^2,\,\, (x,y)\in \Omega_d,$ with Dirichlet boundary conditions. Here $ \Omega_d=\Pi_{j=1}^d]-a_j,a_j[\times \mathbb R_y^d$. The spectral properties of two operators $H$ and $H(\epsilon)$ are investigated. For $\epsilon$ small enough, we study the effect of the slowly varying potential $V(\epsilon x,\epsilon y)$. In particular, we derive asymptotic trace formula and we give an asymptotic expansion in powers of $\epsilon$ of the spectral shift function corresponding to $(H(\epsilon), H)$.