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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Mots clés : Spectral theory, Bochner Laplacian, Semiclassical limit, Magnetic Laplacian
@article{CML_2022__14_1_65_0,
author = {Morin, L\'eo},
title = {Review on {Spectral} asymptotics for the semiclassical {Bochner} {Laplacian} of a line bundle},
journal = {Confluentes Mathematici},
pages = {65--79},
publisher = {Institut Camille Jordan},
volume = {14},
number = {1},
year = {2022},
doi = {10.5802/cml.83},
language = {en},
url = {http://www.numdam.org/articles/10.5802/cml.83/}
}
TY - JOUR AU - Morin, Léo TI - Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle JO - Confluentes Mathematici PY - 2022 SP - 65 EP - 79 VL - 14 IS - 1 PB - Institut Camille Jordan UR - http://www.numdam.org/articles/10.5802/cml.83/ DO - 10.5802/cml.83 LA - en ID - CML_2022__14_1_65_0 ER -
%0 Journal Article %A Morin, Léo %T Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle %J Confluentes Mathematici %D 2022 %P 65-79 %V 14 %N 1 %I Institut Camille Jordan %U http://www.numdam.org/articles/10.5802/cml.83/ %R 10.5802/cml.83 %G en %F CML_2022__14_1_65_0
Morin, Léo. Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle. Confluentes Mathematici, Tome 14 (2022) no. 1, pp. 65-79. doi : 10.5802/cml.83. http://www.numdam.org/articles/10.5802/cml.83/
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