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Spectral analysis of the magnetic Laplacian

Jean-Philippe Miqueu
(IRMAR, Universite de Rennes 1)
Torsdag, 30 april, 2015, at 16:15-17:15, in Aud. D3 (1531-215)

The spectral theory of the Schrödinger equation with magnetic field generates a lot of interest. The aim of its study in the semiclassical limit ($ h \rightarrow 0$) is to understand the link between classical mechanics and quantum mechanics. In this area, most of the work concerns the case of nonvanishing magnetic fields.

This talk will be devoted to the spectral analysis of a self-adjoint realization of the operator :

$P_{h,A}=  (-ih\nabla + A)^2  = \sum_{j=1}^2 (-ih \partial_{x_j} + A_j)^2$

in the semiclassical regime, where the magnetic field $B =\partial_{x_1} A_2 - \partial_{x_1} A_1$    (with ${\mathbf A} \in C^{\infty}({\mathbb R}^2)$ vanishes along a regular curve. We will consider a bounded and simply connected domain of ${\mathbb R}^2$ with smooth boundary.

Such a cancellation changes the structure of the semiclassical limit. The aim is to explore the hierarchy of the model operators which will appear. We will be especially interested in the first asymptotic term of the lowest eigenvalue $\lambda_1(h)$ when the parameter $h$ goes to $0$.

Kontaktperson: Søren Fournais