Spectral shift function for slowly varying perturbation of periodic Schrödinger operators - Dimassi, Mouez; Zerzeri, Maher (CUBO, A Mathematical Journal, 2012)
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 = âˆ’âˆ† + V(ð‘¥)), where ðœ‘(ð‘¥) âˆˆ âˆâˆž(â„n, â„) is a decreasing function, O(|ð‘¥|âˆ’Î´ ) for some Î´ > n and â„Ž is a small positive parameter. Here the potential V is real, smooth and periodic with respect to a lattice Î“ in â„n. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for P(â„Ž).