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

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$W(x)$ispositive,$W(x)\sim w_0(\frac{x}{|x|})|x|^{-\delta}$nearinfinityforsome$\delta>n$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 $\mathrm{\Gamma }$ in ${\mathbb{R}}^n$.