Spectral shift function for perturbed periodic Schroedinger operators. The large-coupling constant limit case

Abstract

In the large coupling constant limit, we obtain an asymptotic expansion in powers of ${\mu }^{-\frac{1}{\delta }}$ of the derivative of the spectral shift function corresponding to the pair $(P_{\mu }=P_0+\mu W(x),P_0=-\Delta +V(x)),$ where $W(x)$ is positive, $W(x)\sim w_0(\frac{x}{|x|})|x{|}^{-\delta }$ near infinity for some $\delta >n$ and $w_0\in {\mathcal{C}}^{\infty }({\mathbb{S}}^{n-1};{\mathbb{R}}_{+}).$ Here ${\mathbb{S}}^{n-1}$ is the unite sphere of the space ${\mathbb{R}}^n$ and $\mu$ is a large parameter. The potential $V$ is real-valued, smooth and periodic with respect to a lattice $\Gamma$ in ${\mathbb{R}}^n$.