Spectral shift function for slowly varying perturbation of periodic Schroedinger operators (1102.2362v2)

Abstract: In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schr\"odinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative of the spectral shift function corresponding to the pair $\big(P(h)=P_0+\phi(hx),P_0=-\Delta+V(x)\big),$ where $\phi(x)\in {\mathcal C}^\infty(\mathbb R^n,\mathbb R)$ is a decreasing function, ${\mathcal O}(|x|^{-\delta})$ for some $\delta>n$ and $h$ is a small positive parameter. Here the potential $V$ is real, smooth and periodic with respect to a lattice $\Gamma$ in ${\mathbb R}^n$. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for $P(h)$.