Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle
Confluentes Mathematici, Volume 14 (2022) no. 1, pp. 65-79.

We first give a short introduction to the Bochner Laplacian on a Riemannian manifold, and explain why it acts locally as a magnetic Laplacian. Then we review recent results on the semiclassical properties of semi-excited spectrum with inhomogeneous magnetic field, including Weyl estimates and eigenvalue asymptotics. These results show under specific assumptions that the spectrum is well described by a familly of operators whose symbols are space-dependent Landau levels. Finally we discuss the strength and limitations of these theorems, in terms of possible crossings between Landau levels.

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Accepted:
Published online:
DOI: 10.5802/cml.83
Classification: 58J50,  35Pxx,  81Q20
Keywords: Spectral theory, Bochner Laplacian, Semiclassical limit, Magnetic Laplacian
Léo Morin 1

1 Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Léo Morin. Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle. Confluentes Mathematici, Volume 14 (2022) no. 1, pp. 65-79. doi : 10.5802/cml.83. https://cml.centre-mersenne.org/articles/10.5802/cml.83/

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