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In the large coupling constant limit, we obtain an asymptotic expansion inpowers of $\mu^{-\frac{1}{\delta}}$ of the derivative of the spectral shiftfunction corresponding to the pair $\big(P_\mu=P_0+\muW(x),P_0=-\Delta+V(x)\big),$ where $W(x)$ is positive, $W(x)\simw_0(\frac{x}{|x|})|x|^{-\delta}$ near infinity for some $\delta>n$ and $w_0\in{\mathcal C}^\infty(\mathbb S^{n-1};\,\mathbb R_+).$ Here $\mathbb S^{n-1}$ isthe unite sphere of the space $\mathbb R^n$ and $\mu$ is a large parameter. Thepotential $V$ is real-valued, smooth and periodic with respect to a lattice$\Gamma$ in ${\mathbb R}^n$.

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