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 $\big(P_\mu=P_0+\mu W(x),P_0=-\Delta+V(x)\big),$ 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 ${\mathbb R}^n$.