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).
@article{1352,
title = {Spectral shift function for slowly varying perturbation of periodic Schr\"{}odinger operators},
journal = {CUBO, A Mathematical Journal},
volume = {14},
year = {2012},
language = {en},
url = {http://dml.mathdoc.fr/item/1352}
}
Dimassi, Mouez; Zerzeri, Maher. Spectral shift function for slowly varying perturbation of periodic Schr¨odinger operators. CUBO, A Mathematical Journal, Tome 14 (2012) . http://gdmltest.u-ga.fr/item/1352/