[Mesures de défaut semi-classiques des laplaciens magnétiques sur les surfaces hyperboliques]
On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three distinct dynamical regimes emerge. In the low-energy regime, we show that any invariant measure of the magnetic flow in phase space can be obtained as a semiclassical measure. At the critical energy level, we establish Quantum Unique Ergodicity, together with a quantitative rate of convergence of eigenfunctions to the Liouville measure. In the high-energy regime, we prove a Shnirelman-type result: a density-one subsequence of eigenfunctions becomes equidistributed with respect to the Liouville measure.
Sur une surface hyperbolique compacte, nous étudions les mesures de défaut semi-classiques associées au laplacien magnétique en présence d’un champ magnétique constant. Selon le niveau d’énergie où les fonctions propres se concentrent, trois régimes dynamiques distincts apparaissent. Dans le régime de basse énergie, nous montrons que toute mesure invariante du flot magnétique dans l’espace des phases peut être obtenue comme mesure semi-classique. Au niveau d’énergie critique, nous établissons l’unique ergodicité quantique, ainsi qu’un taux de convergence quantitatif des fonctions propres vers la mesure de Liouville. Dans le régime de haute énergie, nous démontrons un résultat de type Shnirelman : une sous-suite de densité $1$ des fonctions propres devient équidistribuée par rapport à la mesure de Liouville.
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Keywords: Semiclassical measures, magnetic Laplacian, hyperbolic surfaces, quantum chaos
Mots-clés : Mesures semi-classiques, laplacien magnétique, surfaces hyperboliques, chaos quantique
Laurent Charles 1 ; Thibault Lefeuvre 2
CC-BY 4.0
@article{JEP_2026__13__593_0,
author = {Laurent Charles and Thibault Lefeuvre},
title = {Semiclassical defect measures of {magnetic~Laplacians} on hyperbolic surfaces},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {593--627},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.333},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.333/}
}
TY - JOUR AU - Laurent Charles AU - Thibault Lefeuvre TI - Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 593 EP - 627 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.333/ DO - 10.5802/jep.333 LA - en ID - JEP_2026__13__593_0 ER -
%0 Journal Article %A Laurent Charles %A Thibault Lefeuvre %T Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 593-627 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.333/ %R 10.5802/jep.333 %G en %F JEP_2026__13__593_0
Laurent Charles; Thibault Lefeuvre. Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 593-627. doi: 10.5802/jep.333
[AM22a] - “Concentration of quasimodes for perturbed harmonic oscillators”, 2022 | arXiv | Zbl
[AM22b] - “Localization and delocalization of eigenmodes of harmonic oscillators”, Proc. Amer. Math. Soc. 150 (2022) no. 5, p. 2195-2208 | DOI | Zbl | MR
[Ana08] - “Entropy and the localization of eigenfunctions”, Ann. of Math. (2) 168 (2008) no. 2, p. 435-475 | DOI | Zbl | MR
[Arn61] - “Some remarks on flows of line elements and frames”, Soviet Math. Dokl. 2 (1961), p. 562-564 | Zbl
[Bur90] - “Horocycle flow on geometrically finite surfaces”, Duke Math. J. 61 (1990) no. 3, p. 779-803 | DOI | Zbl | MR
[CdV79] - “Sur le spectre des opérateurs elliptiques à bicaracteristiques toutes périodiques”, Comment. Math. Helv. 54 (1979), p. 508-522 | DOI | MR | Zbl
[CdV85] - “Ergodicité et fonctions propres du laplacien”, in Séminaire Bony-Sjöstrand-Meyer, 1984–1985, École Polytechnique, Palaiseau, 1985, Exp. No. 13, 8 p. | Numdam | Zbl
[Cha00] - Aspects semi-classiques de la quantification géométrique, Ph. D. Thesis, Université Paris Dauphine-PSL, 2000
[Cha24a] - “Landau levels on a compact manifold”, Ann. H. Lebesgue 7 (2024), p. 69-121 | DOI | Numdam | Zbl | MR
[Cha24b] - “On the spectrum of nondegenerate magnetic Laplacians”, Anal. PDE 17 (2024) no. 6, p. 1907-1952 | DOI | Zbl | MR
[Cha25] - “Resolvents of Bochner Laplacians in the semiclassical limit”, Comm. Partial Differential Equations 50 (2025) no. 7, p. 899-930 | DOI | Zbl | MR
[CL] - “Semiclassical analysis on principal bundles” | arXiv | Zbl
[CL25] - “Semiclassical defect measures of magnetic Laplacians on surfaces II. Variable curvature”, 2025, In preparation
[Com86] - “On the Landau levels on the hyperbolic plane”, Ann. Physics 173 (1986), p. 185-209 | DOI | Zbl | MR
[DJ18] - “Semiclassical measures on hyperbolic surfaces have full support”, Acta Math. 220 (2018) no. 2, p. 297-339 | DOI | Zbl | MR
[DJN22] - “Control of eigenfunctions on surfaces of variable curvature”, J. Amer. Math. Soc. 35 (2022) no. 2, p. 361-465 | DOI | MR | Zbl
[Dya22] - “Around quantum ergodicity”, Ann. Math. Qué. 46 (2022) no. 1, p. 11-26 | DOI | MR | Zbl
[DZ19] - Mathematical theory of scattering resonances, Grad. Stud. Math., vol. 200, American Mathematical Society, Providence, RI, 2019 | DOI | MR | Zbl
[FF03] - “Invariant distributions and time averages for horocycle flows”, Duke Math. J. 119 (2003) no. 3, p. 465-526 | DOI | MR | Zbl
[Fur73] - “The unique ergodicity of the horocycle flow”, in Recent advances in topological dynamics (New Haven, Conn., 1972), Lect. Notes in Math., vol. 318, Springer, Berlin-New York, 1973, p. 95-115 | DOI | Zbl | MR
[Gin96] - “On the existence and non-existence of closed trajectories for some Hamiltonian flows”, Math. Z. 223 (1996) no. 3, p. 397-409 | DOI | Zbl | MR
[GU89] - “Circular symmetry and the trace formula”, Invent. Math. 96 (1989) no. 2, p. 385-423 | MR | DOI | Zbl
[IL94] - “Quantum mechanics and quantum Hall effect on Riemann surfaces”, Nuclear Phys. B 413 (1994) no. 3, p. 735-753 | DOI | Zbl | MR
[KT22] - “Trace formula for the magnetic Laplacian on a compact hyperbolic surface”, Regul. Chaotic Dyn. 27 (2022) no. 4, p. 460-476 | DOI | MR | Zbl
[Lef26] - Microlocal analysis in hyperbolic dynamics and geometry, Cours Spécialisés, vol. 32, Société Mathématique de France, Paris, 2026 | MR
[Lin06] - “Invariant measures and arithmetic unique ergodicity. Appendix by E. Lindenstrauss and D. Rudolph”, Ann. of Math. (2) 163 (2006) no. 1, p. 165-219 | DOI | MR | Zbl
[Mac08] - “Some remarks on quantum limits on Zoll manifolds”, Comm. Partial Differential Equations 33 (2008) no. 6, p. 1137-1146 | DOI | MR | Zbl
[Mac09] - “Semiclassical measures and the Schrödinger flow on Riemannian manifolds”, Nonlinearity 22 (2009) no. 5, p. 1003-1020 | DOI | MR | Zbl
[MM24] - “Semiclassical analysis, geometric representation and quantum ergodicity”, Comm. Math. Phys. 405 (2024) no. 11, article ID 259, 28 pages | DOI | MR
[MR00] - “Quantum unique ergodicity for parabolic maps”, Geom. Funct. Anal. 10 (2000) no. 6, p. 1554-1578 | DOI | MR | Zbl
[MR16] - “Concentration and non-concentration for the Schrödinger evolution on Zoll manifolds”, Comm. Math. Phys. 345 (2016) no. 3, p. 1019-1054 | DOI | Zbl | MR
[MR24] - “Quantum unique ergodicity for magnetic Laplacians on T2”, 2024 | arXiv | Zbl
[Non19] - “Introduction to semiclassical analysis”, Lecture notes, available at https://www.imo.universite-paris-saclay.fr/ stephane.nonnenmacher/enseign/Cours-Semiclassical-Analysis2019-lecture1-8.pdf, 2019
[Pat99] - Geodesic flows, Progress in Math., vol. 180, Birkhäuser Boston, Inc., Boston, MA, 1999 | DOI | MR | Zbl
[Ros06] - “Quantum unique ergodicity for maps on the torus”, Ann. Henri Poincaré 7 (2006) no. 3, p. 447-469 | DOI | MR | Zbl
[RS94] - “The behaviour of eigenstates of arithmetic hyperbolic manifolds”, Comm. Math. Phys. 161 (1994) no. 1, p. 195-213 | DOI | MR | Zbl
[Shn74] - “Ergodic properties of eigenfunctions.”, Uspehi Mat. Nauk 29 (1974) no. 6(180), p. 181-182 | MR | Zbl
[Shn22] - “Statistical properties of eigenfunctions”, Ann. Math. Qué. 46 (2022) no. 1, p. 3–9 | DOI | MR | Zbl
[TP06] - “Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces”, Differential Geom. Appl. 24 (2006) no. 3, p. 288-310 | DOI | MR | Zbl
[UZ93] - “Spectral statistics on Zoll surfaces”, Comm. Math. Phys. 154 (1993) no. 2, p. 313-346 | DOI | MR | Zbl
[Wei77] - “Asymptotics of eigenvalue clusters for the Laplacian plus a potential”, Duke Math. J. 44 (1977), p. 883-892 | DOI | MR | Zbl
[Zel87] - “Uniform distribution of eigenfunctions on compact hyperbolic surfaces”, Duke Math. J. 55 (1987), p. 919-941 | DOI | MR | Zbl
[Zel92] - “On a “quantum chaos” theorem of R. Schrader and M. Taylor”, J. Funct. Anal. 109 (1992) no. 1, p. 1-21 | DOI | MR | Zbl
[Zel97] - “Fine structure of Zoll spectra”, J. Funct. Anal. 143 (1997) no. 2, p. 415-460 | DOI | MR | Zbl
[Zel15] - “Gaussian beams on Zoll manifolds and maximally degenerate Laplacians”, in Spectral theory and partial differential equations (Los Angeles, CA, June 2013), American Mathematical Society, Providence, RI, 2015, p. 169-198 | DOI | MR | Zbl
[Zwo12] - Semiclassical analysis, Grad. Stud. Math., vol. 138, American Mathematical Society, Providence, RI, 2012 | DOI | MR | Zbl
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