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Resonances for slowly varying perturbations of a periodic Schrödinger operator. (English) Zbl 1025.81016

From the author’s abstract: We study the resonances of the operator \(P(h) = -\Delta_x + V(x) + \varphi(hx)\). Here \(V\) is a periodic potential, \(\varphi\) a decreasing perturbation and \(h\) a small positive constant. We prove the existence of shape resonances near the edges of the spectral bands of \(P_0 = -\Delta_x + V(x)\), and we give its asymptotic expansions in powers of \(h^{\frac 12}\).

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35Q40 PDEs in connection with quantum mechanics
35B34 Resonance in context of PDEs
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