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Comportement semi-classique pour l’opérateur de Schrödinger à potentiel périodique. (Semi-classical behaviour of the Schrödinger operator with periodic potential). (French) Zbl 0662.35023
The Schrödinger operator with a periodic potential is considered. Let V be a smooth periodic function, we study the semi-classical behavior for a continuum spectrum of \(-h^ 2\Delta +V\) (h\(\to 0)\). We are interested in localization and width of bands. We give the interaction matrix up to an exponentially small error, measured by Agmon’s distance between the wells. A detailed investigation of the spectrum is made for the case where V has one nondegenerate minima per unit cell. We also investigate the spectral properties of \(-h^ 2\Delta +V+\Delta V,\) where \(\Delta\) V is a smooth positive perturbation with compact support.

MSC:
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
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