Magnetic wells in dimension three

Yuri Kordyukov
Salle
Rennes
Date et heure
-
Workshop - Magnetic fields and semi-classical analysis

We will discuss semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms à la Birkhoff. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional $\hbar$-pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of $\hbar^{\frac1 2}$.

This is joint work with B. Helffer, N. Raymond and S. Vu Ngoc.

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