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# Perturbations à oscillations lentes de l'opérateur de Schrödinger périodique.

Abstract : We study the Schrödinger operator $H_{\alpha}=-\frac{\der^2}{\der x^2}+V(x)+W(x^{\alpha})$ in $L_2(\R_+)$, with a generic periodic potential $V$. We suppose that $W$ is periodic and $\alpha\in(0,1)$ so that the perturbation $W(x^{\alpha})$ is asymptotically slowly oscillating. We use two approaches for the asymptotic study of the solutions of the associated eigenvalue equation.\\ The first method is developed by Simon--Zhu and based on periodic approximations. We give an explicit formula for the integrated density of states for $H_{\alpha}$. Then we prove existence and give a formula for the Lyapounov exponent for almost all energies. We obtain a description of the exceptional set of energies containing the singular continuous spectrum of $H_{\alpha}$.\\ The second method is new and uses quasiperiodic approximations instead of periodic ones. We approach the resolvent of $H_{\alpha}$ by the resolvents of the quasiperiodic operators $H_{z,\eps}=-\frac{\der^2}{\der x^2}+V(x)+W(\eps x+z)$ for some parameters $z$ and $\eps$. In order to use the approximate resolvent method for $H_{\alpha}$, we also study the solutions of the eigenvalue equation for $H_{z,\eps}$ using Fedotov--Klopp's complex WKB method. We obtain the asymptotics of the solutions and of the monodromy matrices as $\eps$ goes to zero. Under the condition $\alpha>\frac{1}{2}$, we construct solutions of the eigenvalue equation associated to $H_{\alpha}$ having simple asymptotics in $x$ on large intervals. Then by studying the associated transfer matrices, we obtain a new, more precise than the previous one, description of the exceptional set of energies.\\
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https://theses.hal.science/tel-00714033
Contributor : Asya Metelkina Connect in order to contact the contributor
Submitted on : Tuesday, July 3, 2012 - 11:43:25 AM
Last modification on : Wednesday, October 27, 2021 - 2:52:29 PM
Long-term archiving on: : Thursday, October 4, 2012 - 2:54:11 AM

### Identifiers

• HAL Id : tel-00714033, version 1

### Citation

Asya Metelkina. Perturbations à oscillations lentes de l'opérateur de Schrödinger périodique.. Equations aux dérivées partielles [math.AP]. Université Paris-Nord - Paris XIII, 2011. Français. ⟨NNT : ⟩. ⟨tel-00714033⟩

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