Dimassi, Mouez; Zerzeri, Maher Spectral shift function for slowly varying perturbation of periodic Schrödinger operators. (English) Zbl 1258.81036 Cubo 14, No. 1, 29-47 (2012). Summary: 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 \(\left(P(h) =P_0+\varphi(hx),P_0=-\Delta +V(x)\right)\), where \(\varphi(x)\in C^\infty(\mathbb R^n,\mathbb R)\) is a decreasing function, \(\mathcal O(|x|^{-\delta})\) for some \(\delta>n\) and \(h\) is a small positive parameter. Here the potential \(V\) is real, smooth and periodic with respect to a lattice \(\Gamma\) in \(\mathbb R^n\). To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption theorem for \(P(h)\). Cited in 1 Document MSC: 81Q15 Perturbation theories for operators and differential equations in quantum theory 35J10 Schrödinger operator, Schrödinger equation 35P20 Asymptotic distributions of eigenvalues in context of PDEs 47N50 Applications of operator theory in the physical sciences 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics PDFBibTeX XMLCite \textit{M. Dimassi} and \textit{M. Zerzeri}, Cubo 14, No. 1, 29--47 (2012; Zbl 1258.81036) Full Text: DOI arXiv