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.
CC-BY-NC-ND 4.0
@article{CML_2022__14_1_65_0,
author = {L\'eo Morin},
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 = {https://cml.centre-mersenne.org/articles/10.5802/cml.83/}
}
TY - JOUR AU - Léo Morin 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 - https://cml.centre-mersenne.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 Léo Morin %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 https://cml.centre-mersenne.org/articles/10.5802/cml.83/ %R 10.5802/cml.83 %G en %F CML_2022__14_1_65_0
Léo Morin. 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. https://cml.centre-mersenne.org/articles/10.5802/cml.83/
[1] Lectures on exponential decay of solutions of second order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators, Princeton Legacy Library, Princeton University Press, 1982 no. 29
[2] Almost complex structures and geometric quantization, Mathematical Research Letters, Volume 3 (1996) no. 6, pp. 845-861 | DOI | MR | Zbl
[3] Landau levels on compact manifolds (2020) (arXiv: 2012.14190)
[4] On the spectrum of non degenerate magnetic Laplacian (2021) (arXiv: 2109.05508)
[5] Schrödinger Operators, Theoretical Mathematical Physics, Springer, Berlin, Heidelberg, 1987 | DOI
[6] Champs magnétiques et inégalités de Morse pour la d"-cohomologie, Annales de l’Institut Fourier, Volume 35 (1985) no. 4, pp. 189-229 | DOI | Numdam | Zbl
[7] Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston Inc., 2010 no. 77 | DOI
[8] The Laplace operator on the n-th tensor power of a line bundle: eigenvalues which are uniformly bounded in n, Asymptotic Analysis, Volume 1 (1988) no. 2, pp. 105-113 | DOI | MR | Zbl
[9] Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: The case of discrete wells, Spectral Theory and Geometric Analysis, Volume 535 (2011), pp. 55-78 | DOI | Zbl
[10] Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator. II The case of degenerate wells, Communications in Partial Differential Equations, Volume 37 (2012) no. 4 | Zbl
[11] , Geometric Methods in Physics (Trends in Mathematics) (2014), pp. 259-278 | DOI | Zbl
[12] Magnetic wells in dimension three, Analysis and PDE, Volume 9 (2016) no. 7, pp. 1575-1608 | DOI | MR | Zbl
[13] Semiclassical Analysis for the Ground State Energy of a Schrödinger Operator with Magnetic Wells, Journal of Functional Analysis, Volume 138 (1996), pp. 40-81 | DOI | Zbl
[14] Semiclassical eigenvalue asymptotics for the Bochner Laplacian of a positive line bundle on a symplectic manifold (2019) (arXiv: 1908.01756v1)
[15] Berezin-Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian (2020) (arXiv: 2012.14198)
[16] Semiclassical spectral analysis of the Bochner-Schrödinger operator on symplectic manifolds of bounded geometry, Analysis and Mathematical Physics, Volume 12 (2022) no. 22 | Zbl
[17] Trace formula for the magnetic Laplacian, Russian Mathematical Surveys, Volume 74 (2019) no. 2 | DOI | MR
[18] The Spin-c Dirac operator on high tensor powers of a line bundle, Mathematische Zeitschrift (2002) no. 240, pp. 651-664 | DOI | MR | Zbl
[19] Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface (2018) (arXiv: 1811.00992)
[20] On the spectral theory of the Schrödinger operator with electromagnetic potential, Adv. Part. Diff. Equat., 5 (1994) no. 5, pp. 298-390
[21] A Semiclassical Birkhoff Normal Form for Symplectic Magnetic Wells (2019) (arXiv:1907.03493. To appear in : Journal of Spectral Theory)
[22] A semiclassical Birkhoff normal form for constant-rank magnetic fields (2020) (arXiv:2005.09386)
[23] Bound States of the Magnetic Schrödinger Operator, Tracts in Mathematics, European Mathematical Society, 2017 no. 27 | DOI
[24] Geometry and Spectrum in 2D Magnetic wells, Annales de l’Institut Fourier, Volume 65 (2015) no. 1, pp. 137-169 | DOI | Numdam | MR | Zbl
Cité par Sources :