Spectral shift function for slowly varying perturbation of periodic Schrödinger operators
- Mouez Dimassi dimassi@math.univ-paris13.fr
- Maher Zerzeri zerzeri@math.univ-paris13.fr
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DOI:
https://doi.org/10.4067/S0719-06462012000100004Abstract
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‘¥) ∈ âˆâˆž(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).