Spectral shift function for slowly varying perturbation of periodic Schrödinger operators
Author
Dimassi, Mouez
Zerzeri, Maher
Full text
https://revistas.ufro.cl/ojs/index.php/cubo/article/view/135210.4067/S0719-06462012000100004
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 ℎ of the derivative of the spectral shift function corresponding to the pair (P(ℎ) = P0 + 𝜑(ℎ𝑥), P0 = −∆ + V(𝑥)), where 𝜑(𝑥) ∈ ∁∞(ℝn, ℝ) is a decreasing function, O(|𝑥|−δ ) for some δ > n and ℎ is a small positive parameter. Here the potential V is real, smooth and periodic with respect to a lattice Γ in ℝn. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for P(ℎ).