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Spectral shift function for slowly varying perturbation of periodic Schrödinger operators. (English) Zbl 1258.81036

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 \(P(h)\).

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
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