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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 $P\left(h\right)=-{{\Delta }}_{x}+V\left(x\right)+\varphi \left(hx\right)$. 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 ${P}_{0}=-{{\Delta }}_{x}+V\left(x\right)$, and we give its asymptotic expansions in powers of ${h}^{\frac{1}{2}}$.
##### MSC:
 81Q20 Semi-classical techniques in quantum theory, including WKB and Maslov methods 35Q40 PDEs in connection with quantum mechanics 35B34 Resonances in solutions of PDE
##### Keywords:
resonances; semi-classical limit