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

Keywords:

resonances; semi-classical limit
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