Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle

Abstract

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A semiclassical Birkhoff normal form for constant-rank magnetic fields

“…Remark 3.5. U − h is a Fourier integral operator quantizing the symplectomorphism ˜ 1 ; see [Martinez 2002;Zworski 2012]. In particular, if…”

“…The magnetic field is the 2-form B = dA. The spectral theory of the magnetic Laplacian has given rise to many investigations, and appeared to have very various behaviors according to the variations of B and the geometry of M. We refer to the books and review [Helffer and Kordyukov 2014;Fournais and Helffer 2010;Raymond 2017] for a description of these works. Here we focus on the Dirichlet realization of L − h , and we give a description of semiexcited states, eigenvalues of order O( − h) in the semiclassical limit − h → 0.…”

“…The classical Birkhoff normal form has a long story in physics and goes back to [Delaunay 1860;Lindstedt 1883]. This formal normal form was the starting point of a lot of studies on stability near equilibrium, and KAM theory (after [Kolmogorov 1954;Arnold 1963;Moser 1962]). The name of this normal form comes from [Birkhoff 1927;Gustavson 1966].…”

“…The name of this normal form comes from [Birkhoff 1927;Gustavson 1966]. We refer to the books [Moser 1968;Hofer and Zehnder 1994] for precise statements. Our approach here relies on a quantization.…”

“…Our approach here relies on a quantization. Physicists and quantum chemists already noticed in the 1980s that a quantum analogue of the Birkhoff normal form could be used to compute energies of molecules [Delos et al 1983;Jaffé and Reinhardt 1982;Marcus 1985;Shirts and Reinhardt 1982]. Joyeux and Sugny [2002] also used such techniques to describe the dynamics of excited states.…”