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

Abstract

In the large coupling constant limit, we obtain an asymptotic expansion in powers of μ^−1/δ of the derivative of the spectral shift function corresponding to the pair (P_μ=P₀+μW(x),P₀=−Δ_x+V(x)), where W(x) is positive, W(x)~w₀(x/|x|)|x|^−δ near infinity for some δ>n and w₀∈C^∞(Sⁿ⁻¹;R₊). Here Sⁿ⁻¹ is the unite sphere of the space Rⁿ and μ is a large parameter. The potential V is real-valued, smooth and periodic with respect to a lattice Γ in Rⁿ.