Geometry and Spectrum in 2D Magnetic Wells

Raymond, Nicolas; Vũ Ngọc, San

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 137-169.

Cet article est consacré à la mécanique classique et l’analyse spectrale d’un hamiltonien purement magnétique dans ${ℝ}^2$. On démontre que la dynamique et la théorie spectrale semi-classique peuvent être traitées par une forme normale de Birkhoff, et ainsi réduites à l’étude d’une famille d’hamiltoniens à un degré de liberté. Corollairement, on obtient une extension de résultats récents de Helffer et Kordyukov à de plus hautes énergies.

This paper is devoted to the classical mechanics and spectral analysis of a pure magnetic Hamiltonian in ${ℝ}^2$. It is established that both the dynamics and the semiclassical spectral theory can be treated through a Birkhoff normal form and reduced to the study of a family of one dimensional Hamiltonians. As a corollary, recent results by Helffer-Kordyukov are extended to higher energies.

Classification : 81Q20, 35Pxx, 35S05, 70Hxx, 37Jxx
Keywords: magnetic field, normal form, spectral theory, semiclassical limit, Hamiltonian flow, microlocal analysis
Mot clés : champ magnétique, forme normale, théorie spectrale, limite semi-classique, flot hamiltonien, analyse microlocale

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