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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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##### References:
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