Semiclassical analysis of the magnetic Laplacian on hyperbolic surfaces
[Analyse semiclassique du Laplacien magnétique sur les surfaces hyperboliques]
Thibault Lefeuvre1
1Université de Paris and Sorbonne Université, CNRS, IMJ-PRG, F-75006 Paris, France.
Journées équations aux dérivées partielles (2025), Exposé no. 4, 10 p.

The magnetic Laplacian on hyperbolic surfaces provides a rich analytic framework in which a variety of quantum phenomena emerge. The present note, written for the Proceedings of the Journées EDP 2025, is a concise overview of the main results obtained in [8, 9, 10].

Le Laplacien magnétique sur les surfaces hyperboliques fournit un cadre analytique remarquable, dans lequel un certain nombre de phénomènes quantiques émergent. La présente note, écrite pour les Proceedings of the Journées EDP 2025, propose de résumer les principaux résultats obtenus dans [8, 9, 10].

Publié le :
DOI : 10.5802/jedp.695
Classification : 35P20, 58J50, 35S05, 35J10, 37C40, 53C22
Keywords: Semiclassical analysis, hyperbolic surfaces, magnetic Laplacian, dynamical systems, spectral theory

Thibault Lefeuvre  1

1 Université de Paris and Sorbonne Université, CNRS, IMJ-PRG, F-75006 Paris, France. 
Thibault Lefeuvre. Semiclassical analysis of the magnetic Laplacian on hyperbolic surfaces. Journées équations aux dérivées partielles (2025), Exposé no. 4, 10 p.. doi: 10.5802/jedp.695
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[1] Nalini Anantharaman Entropy and the localization of eigenfunctions, Ann. Math., Volume 168 (2008) no. 2, pp. 435-475 | DOI | Zbl

[2] Nalini Anantharaman; Stéphane Nonnenmacher Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier, Volume 57 (2007) no. 7, pp. 2465-2523 | DOI | Numdam | Zbl | MR

[3] Vojislav G. Avakumović Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z., Volume 65 (1956), pp. 327-344 | DOI | Zbl | MR

[4] Pierre H. Bérard On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | DOI | Zbl | MR

[5] Yannick Bonthonneau The theta function and the Weyl law on manifolds without conjugate points, Doc. Math., Volume 22 (2017), pp. 1275-1283 | Zbl | DOI | MR

[6] Marc Burger Horocycle flow on geometrically finite surfaces, Duke Math. J., Volume 61 (1990) no. 3, pp. 779-803 | DOI | Zbl | MR

[7] Yaiza Canzani; Jeffrey Galkowski Improvements for eigenfunction averages: an application of geodesic beams, J. Differ. Geom., Volume 124 (2023) no. 3, pp. 443-522 | Zbl | MR

[8] Ambre Chabert; Thibault Lefeuvre Improved L ∞ -bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces (in preparation) | Zbl

[9] Laurent Charles; Thibault Lefeuvre Semiclassical defect measures for magnetic Laplacians on surfaces near zero energy (in preparation) | Zbl

[10] Laurent Charles; Thibault Lefeuvre Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces, J. Éc. Polytech., Math., Volume 13 (2026), pp. 593-627 | Zbl | DOI | MR

[11] Harold Donnelly Bounds of eigenfunctions of the Laplacian on compact Riemannian manifolds, J. Funct. Anal., Volume 187 (2001) no. 1, pp. 247-261 | DOI | Zbl | MR

[12] Semyon Dyatlov Around quantum ergodicity, Ann. Math. Qué., Volume 46 (2022) no. 1, pp. 11-26 | DOI | Zbl | MR

[13] Semyon Dyatlov; Long Jin Semiclassical measures on hyperbolic surfaces have full support, Acta Math., Volume 220 (2018) no. 2, pp. 297-339 | Zbl | DOI | MR

[14] Semyon Dyatlov; Long Jin; Stéphane Nonnenmacher Control of eigenfunctions on surfaces of variable curvature, J. Am. Math. Soc., Volume 35 (2022) no. 2, pp. 361-465 | DOI | Zbl | MR

[15] Harry Furstenberg The unique ergodicity of the horocycle flow, Recent advances in topological dynamics. Proceedings of the conference on topological dynamics, held at Yale University, June 19-23, 1972 (Lecture Notes in Mathematics), Volume 318, Springer, 1973, pp. 95-115 | Zbl | MR

[16] Lars Hörmander The spectral function of an elliptic operator, Acta Math., Volume 121 (1968) no. 1, pp. 193-218 | Zbl | DOI | MR

[17] Maxime Ingremeau; Martin Vogel Improved L ∞ bounds for eigenfunctions under random perturbations in negative curvature (2024) | arXiv | DOI | Zbl

[18] Henryk Iwaniec; Peter Sarnak L ∞ norms of eigenfunctions of arithmetic surfaces, Ann. Math., Volume 141 (1995), pp. 301-320 | DOI | Zbl

[19] Boris M. Levitan On the asymptotic behavior of the spectral function of a self-adjoint differential operator of second order, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 16 (1952), pp. 325-352 | Zbl | MR

[20] Léo Morin; Gabriel Rivière Quantum unique ergodicity for magnetic Laplacians on T2 (2025) | arXiv | Zbl

[21] Peter Sarnak Arithmetic quantum chaos, The Schur lectures (1992), Tel Aviv (Israel Mathematical Conference Proceedings), Volume 8, American Mathematical Society, 1995, pp. 183-236 | Zbl

[22] Alan Weinstein Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J., Volume 44 (1977), pp. 883-892 | DOI | Zbl | MR

[23] Steve Zelditch Fine structure of Zoll spectra, J. Funct. Anal., Volume 143 (1997) no. 2, pp. 415-460 | DOI | Zbl | MR

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