Resumo

DIMASSI, Mouez  e  ZERZERI, Maher. Spectral shift function for slowly varying perturbation of periodic Schrödinger operators. Cubo [online]. 2012, vol.14, n.1, pp. 29-47. ISSN 0719-0646.  doi: 10.4067/S0719-06462012000100004.

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schrödinger 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 = -Δ+ v(x)) ; where 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 T in Rⁿ. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for P(h).

Palavras-chave : Periodic Schrödinger operator; spectral shift function; asymptotic expansions; limiting absorption theorem.