# Etude de perturbations adiabatiques de l'équation de Schrödinger périodique

Abstract : This work is devoted to the study of adiabatic perturbations of the periodic one-dimensional Schrödinger equation. We consider the operator $H_(\varphi,\varepsilon)=-\Delta+[V(x)+W(\varepsilon x+\varphi)]$ where $V$ is periodic, $W$ tends to zero as $x$ tends to infinity, $\varepsilon$ and $\varphi$ are real. We deal with the adiabatic limit where $\varepsilon$ is a small parameter. We are interested in the eigenvalues of $H_(\varphi,\varepsilon)$ in the gaps of the periodic operator $-\Delta+V$; under suitable assumptions for $W$, these eigenvalues are created by the extrema of $W$. If there is only one extremum, we prove that these eigenvalues oscillate around some quantized energies, given by a Bohr-Sommerfeld quantization rule. The amplitude of the oscillations is exponentially small and determined by a tunneling coefficient. If there are two extrema, each creates a sequence of eigenvalues; these can be resonant. In that case, we highlight a splitting phenomenon of the eigenvalues; this phenomenon is analogous to the well-known splitting for the double well.
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https://tel.archives-ouvertes.fr/tel-00007868
Contributor : Magali Marx <>
Submitted on : Thursday, December 30, 2004 - 9:58:20 AM
Last modification on : Tuesday, October 20, 2020 - 3:56:29 PM
Long-term archiving on: : Thursday, September 13, 2012 - 1:55:19 PM

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• HAL Id : tel-00007868, version 1

### Citation

Magali Marx. Etude de perturbations adiabatiques de l'équation de Schrödinger périodique. Mathématiques [math]. Université Paris-Nord - Paris XIII, 2004. Français. ⟨tel-00007868⟩

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