Localization and delocalization of eigenmodes of harmonic oscillators
HTML articles powered by AMS MathViewer
- by Víctor Arnaiz and Fabricio Macià PDF
- Proc. Amer. Math. Soc. 150 (2022), 2195-2208 Request permission
Abstract:
We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems of coupled quantum harmonic oscillators with arbitrary frequencies. The structure of the set of semi-classical measures turns out to depend strongly on the arithmetic relations between frequencies of each decoupled oscillator. In particular, we show that as soon as these frequencies are not rational multiples of a fixed fundamental frequency, the set of semi-classical measures is not convex and therefore, infinitely many measures that are invariant under the classical harmonic oscillator are not semi-classical measures.References
- Nalini Anantharaman, Entropy and the localization of eigenfunctions, Ann. of Math. (2) 168 (2008), no. 2, 435–475. MR 2434883, DOI 10.4007/annals.2008.168.435
- Nalini Anantharaman, Clotilde Fermanian-Kammerer, and Fabricio Macià, Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures, Amer. J. Math. 137 (2015), no. 3, 577–638. MR 3357117, DOI 10.1353/ajm.2015.0020
- Nalini Anantharaman, Matthieu Léautaud, and Fabricio Macià, Wigner measures and observability for the Schrödinger equation on the disk, Invent. Math. 206 (2016), no. 2, 485–599. MR 3570298, DOI 10.1007/s00222-016-0658-4
- Nalini Anantharaman and Stéphane Nonnenmacher, Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2465–2523 (English, with English and French summaries). Festival Yves Colin de Verdière. MR 2394549, DOI 10.5802/aif.2340
- V. Arnaiz, Semiclassical measures and asymptotic distribution of eigenvalues for quantum KAM systems, Ph.D. dissertation, Universidad Autónoma de Madrid, 2018.
- Víctor Arnaiz, Spectral stability and semiclassical measures for renormalized KAM systems, Nonlinearity 33 (2020), no. 6, 2562–2591. MR 4105369, DOI 10.1088/1361-6544/ab7724
- Daniel Azagra and Fabricio Macià, Concentration of symmetric eigenfunctions, Nonlinear Anal. 73 (2010), no. 3, 683–688. MR 2653740, DOI 10.1016/j.na.2010.03.056
- Jean Bourgain and Elon Lindenstrauss, Entropy of quantum limits, Comm. Math. Phys. 233 (2003), no. 1, 153–171. MR 1957735, DOI 10.1007/s00220-002-0770-8
- Mihajlo Cekić, Bogdan Georgiev, and Mayukh Mukherjee, Polyhedral billiards, eigenfunction concentration and almost periodic control, Comm. Math. Phys. 377 (2020), no. 3, 2451–2487. MR 4121624, DOI 10.1007/s00220-020-03741-0
- Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), no. 3, 497–502 (French, with English summary). MR 818831, DOI 10.1007/BF01209296
- Monique Combescure and Didier Robert, Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics, Springer, Dordrecht, 2012. MR 2952171, DOI 10.1007/978-94-007-0196-0
- S. De Bièvre, J.-C. Houard, and M. Irac-Astaud, Wave packets localized on closed classical trajectories, Differential equations with applications to mathematical physics, Math. Sci. Engrg., vol. 192, Academic Press, Boston, MA, 1993, pp. 25–32. MR 1207145, DOI 10.1016/S0076-5392(08)62369-3
- Mouez Dimassi and Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. MR 1735654, DOI 10.1017/CBO9780511662195
- Semyon Dyatlov and Long Jin, Semiclassical measures on hyperbolic surfaces have full support, Acta Math. 220 (2018), no. 2, 297–339. MR 3849286, DOI 10.4310/ACTA.2018.v220.n2.a3
- Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
- P. Gérard, Mesures semi-classiques et ondes de Bloch, Séminaire sur les Équations aux Dérivées Partielles, 1990–1991, École Polytech., Palaiseau, 1991, pp. Exp. No. XVI, 19 (French). MR 1131589
- S. Gomes, KAM Hamiltonians are not quantum ergodic, arXiv:1811.07718, 2018.
- S. Gomes and A. Hassell, Semiclassical scarring on tori in KAM Hamiltonian systems, arXiv:1811.11346, 2018.
- Eugene Gutkin, Billiards in polygons, Phys. D 19 (1986), no. 3, 311–333. MR 844706, DOI 10.1016/0167-2789(86)90062-X
- Andrew Hassell, Ergodic billiards that are not quantum unique ergodic, Ann. of Math. (2) 171 (2010), no. 1, 605–618. With an appendix by the author and Luc Hillairet. MR 2630052, DOI 10.4007/annals.2010.171.605
- Andrew Hassell, Luc Hillairet, and Jeremy Marzuola, Eigenfunction concentration for polygonal billiards, Comm. Partial Differential Equations 34 (2009), no. 4-6, 475–485. MR 2530705, DOI 10.1080/03605300902768909
- Dmitry Jakobson, Quantum limits on flat tori, Ann. of Math. (2) 145 (1997), no. 2, 235–266. MR 1441877, DOI 10.2307/2951815
- Dmitry Jakobson and Steve Zelditch, Classical limits of eigenfunctions for some completely integrable systems, Emerging applications of number theory (Minneapolis, MN, 1996) IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 329–354. MR 1691539, DOI 10.1007/978-1-4612-1544-8_{1}3
- Elon Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), no. 1, 165–219. MR 2195133, DOI 10.4007/annals.2006.163.165
- Fabricio Macià, Some remarks on quantum limits on Zoll manifolds, Comm. Partial Differential Equations 33 (2008), no. 4-6, 1137–1146. MR 2424392, DOI 10.1080/03605300802038601
- Fabricio Macià and Gabriel Rivière, Concentration and non-concentration for the Schrödinger evolution on Zoll manifolds, Comm. Math. Phys. 345 (2016), no. 3, 1019–1054. MR 3519588, DOI 10.1007/s00220-015-2504-8
- Jens Marklof and Zeév Rudnick, Almost all eigenfunctions of a rational polygon are uniformly distributed, J. Spectr. Theory 2 (2012), no. 1, 107–113. MR 2879311, DOI 10.4171/JST/23
- Daisy Ojeda-Valencia and Carlos Villegas-Blas, On limiting eigenvalue distribution theorems in semiclassical analysis, Spectral analysis of quantum Hamiltonians, Oper. Theory Adv. Appl., vol. 224, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 221–252. MR 2962862, DOI 10.1007/978-3-0348-0414-1_{1}1
- Zeév Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195–213. MR 1266075, DOI 10.1007/BF02099418
- A. I. Šnirel′man, Ergodic properties of eigenfunctions, Uspehi Mat. Nauk 29 (1974), no. 6(180), 181–182 (Russian). MR 0402834
- E. Studnia, Quantum limits for harmonic oscillator, arXiv:1905.07763, 2019.
- John A. Toth, On the quantum expected values of integrable metric forms, J. Differential Geom. 52 (1999), no. 2, 327–374. MR 1758299
- Peter Woit, Quantum theory, groups and representations, Springer, Cham, 2017. An introduction. MR 3726869, DOI 10.1007/978-3-319-64612-1
- Jared Wunsch, Non-concentration of quasimodes for integrable systems, Comm. Partial Differential Equations 37 (2012), no. 8, 1430–1444. MR 2957546, DOI 10.1080/03605302.2011.626102
- Steven Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), no. 4, 919–941. MR 916129, DOI 10.1215/S0012-7094-87-05546-3
- Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218, DOI 10.1090/gsm/138
Additional Information
- Víctor Arnaiz
- Affiliation: Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405 Orsay, France; and Laboratoire de Mathematiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2 rue de la Houssiniere, 44322 Nantes Cedex 03, France
- ORCID: 0000-0002-3390-0596
- Email: victor.arnaiz@universite-paris-saclay.fr
- Fabricio Macià
- Affiliation: M$^2$ASAI, Universidad Politécnica de Madrid, ETSI Navales, Avda. de la Memoria, 4, 28040 Madrid, Spain
- ORCID: 0000-0002-0221-2889
- Email: fabricio.macia@upm.es
- Received by editor(s): January 27, 2021
- Received by editor(s) in revised form: July 21, 2021, and July 26, 2021
- Published electronically: February 7, 2022
- Additional Notes: The first author was supported by a predoctoral grant from Fundación La Caixa - Severo Ochoa International Ph.D. Program at the Instituto de Ciencias Matemáticas (ICMAT-CSIC-UAM-UC3M-UCM), and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 725967). Both authors were partially supported by grant MTM2017-85934-C3-3-P (MINECO, Spain).
- Communicated by: Tanya Christiansen
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 2195-2208
- MSC (2020): Primary 58J51; Secondary 35P20, 35Q40
- DOI: https://doi.org/10.1090/proc/15767
- MathSciNet review: 4392353