Spectral shift function for slowly varying perturbation of periodic Schrödinger operators

Abstract

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schr¨odinger operators. We give a weak and pointwise asymptotic expansions in powers of â„Ž of the derivative of the spectral shift function corresponding to the pair (P(â„Ž) = P₀ + ðœ‘(â„Žð‘¥), P₀ = −∆ + V(ð‘¥)), where ðœ‘(ð‘¥) ∈ âˆ^∞(â„ⁿ, â„) is a decreasing function, O(|ð‘¥|^−δ ) for some δ > n and â„Ž is a small positive parameter. Here the potential V is real, smooth and periodic with respect to a lattice Γ in â„ⁿ. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for P(â„Ž).