Sur le spectre de l'opérateur de Schrödinger magnétique dans un domaine diédral

Abstract : This thesis analyses the spectrum of magnetic Schrödinger operators with constant magnetic field in dihedral domains. In order to understand how a curved edge influences the first eigenvalue in the semi-classical limit, we have to investigate the bottom of the spectrum of the Schrödinger operator with constant magnetic field on an infinite wedge. Using Fourier transform we reduce to a one-parameter family of operators on the corresponding infinite sector. We exhibit the essential spectrum of the operators on the sector and we show that in certain cases there exist discrete eigenvalues below the essential spectrum. Comparing with singular Sturm-Liouville operators on the half-line, we get upper bounds for the bottom of the spectrum of the magnetic Schrödinger operator on the wedge: For small opening angles and some special orientations of the magnetic field, this quantity is smaller than the spectral quantities coming from the regular case. We finally use these results to investigate the Schrödinger operator with constant magnetic field and small parameter in three-dimensional bounded domains with a curved edge. In order to find asymptotics for the first eigenvalue in the semi-classical limit, we construct quasi-modes near the edge using the eigenfunctions of the parameter problem on the sector. Using a partition of unity depending on whether we are close to the edge or to the regular boundary, we get the first term of the asymptotics for various orientations of the magnetic field and we show that the first eigenvalue can be smaller than in the regular case.
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Submitted on : Friday, November 23, 2012 - 11:00:53 AM
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  • HAL Id : tel-00746794, version 2

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Nicolas Popoff. Sur le spectre de l'opérateur de Schrödinger magnétique dans un domaine diédral. Théorie spectrale [math.SP]. Université Rennes 1, 2012. Français. ⟨tel-00746794v2⟩

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