Strong scarring of logarithmic quasimodes
[Quasimode logarithmique et grosse balafre]
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2307-2347.

Nous considérons un opérateur pseudodifférentiel semiclassique sur une surface compacte, tel que le flot Hamiltonien engendré par son symbole principal possède, à une certaine énergie, une orbite périodique hyperbolique. Pour un paramètre ε>0 arbitrairement petit, nous construisons une famille de quasimodes de cet opérateur, dont la largeur en énergie est d’ordre ε/|log|, mais qui possèdent un poids positif (une «  grosse balafre ») autour de cette orbite périodique. Notre construction procède par un contrôle de l’évolution de paquets d’onde gaussiens jusqu’au temps d’Ehrenfest.

We consider a semiclassical pseudodifferential operator on a compact surface, such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit at some energy. For an arbitrary small ε>0, we construct semiclassical families of quasimodes of this operator, with energy widths of order ε/|log|, and which feature a strong scar along that hyperbolic orbit. Our construction proceeds by controlling the evolution of Gaussian wavepackets up to the Ehrenfest time.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3137
Classification : 35-xx,  58Jxx,  37-xx
Mots clés : analyse semiclassique, quasimode, unique ergodicité quantique, balafre d’orbite périodique
@article{AIF_2017__67_6_2307_0,
     author = {Eswarathasan, Suresh and Nonnenmacher, St\'ephane},
     title = {Strong scarring of logarithmic quasimodes},
     journal = {Annales de l'Institut Fourier},
     pages = {2307--2347},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     doi = {10.5802/aif.3137},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3137/}
}
Eswarathasan, Suresh; Nonnenmacher, Stéphane. Strong scarring of logarithmic quasimodes. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2307-2347. doi : 10.5802/aif.3137. http://archive.numdam.org/articles/10.5802/aif.3137/

[1] Anantharaman, Nalini Entropy and the localization of eigenfunctions, Ann. Math., Volume 168 (2008) no. 2, pp. 435-475 | Article | MR 2434883 | Zbl 1175.35036

[2] Anantharaman, Nalini; Nonnenmacher, Stéphane Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier, Volume 57 (2007) no. 7, pp. 2465-2523 http://aif.cedram.org/item?id=AIF_2007__57_7_2465_0 (Festival Yves Colin de Verdière) | Article | MR 2394549 | Zbl 1145.81033

[3] Babič, V. M.; Lazutkin, Vladimir F. The eigenfunctions which are concentrated near a closed geodesic, Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems (Russian), Izdat. Leningrad. Univ., Leningrad, 1967, pp. 15-25 | MR 0234391

[4] Barnett, Aalexander Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards, Commun. Pure Appl. Math., Volume 59 (2006) no. 10, pp. 1457-1488 | Article | MR 2248896 | Zbl 1133.81022

[5] Borondo, Florentino; de Polavieja, Gonzalo G.; Benito, Rosa M. Scars in groups of eigenfunctions, Hamiltonian mechanics (Toruń, 1993) (NATO Adv. Sci. Inst. Ser. B Phys.), Volume 331, Plenum, New York, 1994, pp. 287-294 | Article | MR 1316687

[6] Brooks, Shimon Logarithmic-scale quasimodes that do not equidistribute, Int. Math. Res. Not. (2015) no. 22, pp. 11934-11960 | MR 3456709 | Zbl 1357.37054

[7] Brooks, Shimon Eisenstein quasimodes and QUE, Ann. Henri Poincaré, Volume 17 (2016) no. 3, pp. 615-643 | Article | MR 3459122 | Zbl 1337.81063

[8] Brooks, Shimon; Lindenstrauss, Elon Joint quasimodes, positive entropy, and quantum unique ergodicity, Invent. Math., Volume 198 (2014) no. 1, pp. 219-259 | Article | MR 3260861 | Zbl 1343.58016

[9] Burq, Nicolas; Zworski, Maciej Geometric control in the presence of a black box, J. Am. Math. Soc., Volume 17 (2004) no. 2, pp. 443-471 | Article | MR 2051618 | Zbl 1050.35058

[10] Christianson, Hans Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., Volume 246 (2007) no. 2, pp. 145-195 corrigendum in ibid. 253 (2010), no. 3, p. 1060-1065 | Article | MR 2321040 | Zbl 1119.58018

[11] Christianson, Hans Quantum monodromy and nonconcentration near a closed semi-hyperbolic orbit, Trans. Am. Math. Soc., Volume 363 (2011) no. 7, pp. 3373-3438 | Article | MR 2775812 | Zbl 1230.58020

[12] Combescure, Monique; Robert, Didier Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptotic Anal., Volume 14 (1997) no. 4, pp. 377-404 | MR 1461126 | Zbl 0894.35026

[13] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, 1999, xii+227 pages | Article | MR 1735654 | Zbl 0926.35002

[14] Duistermaat, Johannes Jisse Oscillatory integrals, Lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math., Volume 27 (1974), pp. 207-281 | Article | MR 0405513 | Zbl 0285.35010

[15] Dyatlov, Semyon; Guillarmou, Colin Microlocal limits of plane waves and Eisenstein functions, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 2, pp. 371-448 | Article | MR 3215926 | Zbl 1297.58007

[16] Dyatlov, Semyon; Jin, Long Semiclassical measures on hyperbolic surfaces have full support (2017) https://arxiv.org/abs/1705.05019 (https://arxiv.org/abs/1705.05019)

[17] Eswarathasan, Suresh; Silberman, Lior Scarring of quasimodes on hyperbolic manifolds (2016) http://https://arxiv.org/abs/1609.04912 (http://https://arxiv.org/abs/1609.04912)

[18] Faure, Frédéric; Nonnenmacher, Stéphane On the maximal scarring for quantum cat map eigenstates, Commun. Math. Phys., Volume 245 (2004) no. 1, pp. 201-214 | Article | MR 2036373 | Zbl 1071.81044

[19] Faure, Frédéric; Nonnenmacher, Stéphane; De Bièvre, Stephan Scarred eigenstates for quantum cat maps of minimal periods, Commun. Math. Phys., Volume 239 (2003) no. 3, pp. 449-492 | Article | MR 2000926 | Zbl 1033.81024

[20] Gérard, Christian; Sjöstrand, Johannes Semiclassical resonances generated by a closed trajectory of hyperbolic type, Commun. Math. Phys., Volume 108 (1987) no. 3, pp. 391-421 | Article | MR 874901 | Zbl 0637.35027

[21] Gérard, Patrick; Leichtnam, Éric Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., Volume 71 (1993) no. 2, pp. 559-607 | Article | MR 1233448 | Zbl 0788.35103

[22] Guillemin, Victor Wave-trace invariants, Duke Math. J., Volume 83 (1996) no. 2, pp. 287-352 | Article | MR 1390650 | Zbl 0858.58051

[23] Guillemin, Victor; Paul, Thierry Some remarks about semiclassical trace invariants and quantum normal forms, Commun. Math. Phys., Volume 294 (2010) no. 1, pp. 1-19 | Article | MR 2575473 | Zbl 1213.58021

[24] Guillemin, Victor; Weinstein, Alan Eigenvalues associated with a closed geodesic, Bull. Am. Math. Soc., Volume 82 (1976) no. 1, pp. 92-94 corrigendum in ibid. 82 (1976), p. 966 | Article | MR 0436227 | Zbl 0317.35071

[25] Hagedorn, George A. Semiclassical quantum mechanics. I. The 0 limit for coherent states, Commun. Math. Phys., Volume 71 (1980) no. 1, pp. 77-93 http://projecteuclid.org/euclid.cmp/1103907396 | Article | MR 556903

[26] Hagedorn, George A.; Joye, Alain Semiclassical dynamics with exponentially small error estimates, Commun. Math. Phys., Volume 207 (1999) no. 2, pp. 439-465 | Article | MR 1724830 | Zbl 1031.81519

[27] Heller, Eric J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits, Phys. Rev. Lett., Volume 53 (1984) no. 16, pp. 1515-1518 | Article | MR 762412

[28] Hörmander, Lars The analysis of linear partial differential operators. III Pseudo-differential operators, Classics in Mathematics, Springer, 2007, viii+525 pages (reprint of the 1994 edition) | Article | MR 2304165 | Zbl 1115.35005

[29] Kaplan, L. Scars in quantum chaotic wave functions, Nonlinearity, Volume 12 (1999) no. 2, p. R1-R40 | Article | MR 1677744 | Zbl 0966.37046

[30] Kaplan, L.; Heller, Eric J. Measuring scars of periodic orbits, Phys. Rev. E, Volume 59 (1999) no. 6, pp. 6609-6628 | Article | MR 1695455

[31] Keller, Joseph B. Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Phys., Volume 4 (1958), pp. 180-188 | Article | MR 0099207 | Zbl 0085.43103

[32] Lindenstrauss, Elon Invariant measures and arithmetic quantum unique ergodicity, Ann. Math., Volume 163 (2006) no. 1, pp. 165-219 | Article | MR 2195133 | Zbl 1104.22015

[33] Maslov, Viktor Theory of perturbations and asymptotic methods, Études Mathématiques, Gauthier-Villars, 1972, 384 pages http://www.maths.ed.ac.uk/~aar/papers/maslovbook.pdf (French translation of Russian text, available at http://www.maths.ed.ac.uk/~aar/papers/maslovbook.pdf)

[34] Nonnenmacher, Stéphane; Voros, André Eigenstate structures around a hyperbolic point, J. Phys. A, Math. Gen., Volume 30 (1997) no. 1, pp. 295-315 | Article | MR 1447118 | Zbl 0922.58026

[35] Ralston, James V. On the construction of quasimodes associated with stable periodic orbits, Commun. Math. Phys., Volume 51 (1976) no. 3, pp. 219-242 http://projecteuclid.org/euclid.cmp/1103900389 | Article | MR 0426057 | Zbl 0333.35066

[36] Rudnick, Zeév; Sarnak, Peter The behaviour of eigenstates of arithmetic hyperbolic manifolds, Commun. Math. Phys., Volume 161 (1994) no. 1, pp. 195-213 http://projecteuclid.org/euclid.cmp/1104269797 | Article | MR 1266075 | Zbl 0836.56043

[37] Schubert, Roman; Vallejos, Raúl O.; Toscano, Fabricio How do wave packets spread? Time evolution on Ehrenfest time scales, J. Phys. A, Math. Theor., Volume 45 (2012) no. 21 (Article ID 215307, 28 pp.) | Article | MR 2925343 | Zbl 1301.81070

[38] Sjöstrand, Johannes Resonances associated to a closed hyperbolic trajectory in dimension 2, Asymptotic Anal., Volume 36 (2003) no. 2, pp. 93-113 | MR 2021528 | Zbl 1060.35096

[39] Šnirelʼman, Alexander I. Ergodic properties of eigenfunctions, Usp. Mat. Nauk, Volume 29 (1974) no. 6(180), p. 181-182 | MR 0402834 | Zbl 0324.58020

[40] Toth, John A. Eigenfunction localization in the quantized rigid body, J. Differ. Geom., Volume 43 (1996) no. 4, pp. 844-858 http://projecteuclid.org/euclid.jdg/1214458534 | Article | MR 1412687 | Zbl 0871.58050

[41] Toth, John A. On the quantum expected values of integrable metric forms, J. Differ. Geom., Volume 52 (1999) no. 2, pp. 327-374 http://projecteuclid.org/euclid.jdg/1214425280 | Article | MR 1758299 | Zbl 0992.53063

[42] Colin de Verdière, Yves Quasi-modes sur les variétés Riemanniennes, Invent. Math., Volume 43 (1977) no. 1, pp. 15-52 | Article | MR 0501196 | Zbl 0449.53040

[43] Colin de Verdière, Yves Ergodicité et fonctions propres du laplacien, Commun. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 http://projecteuclid.org/euclid.cmp/1104114465 | Article | MR 818831 | Zbl 0592.58050

[44] Colin de Verdière, Yves; Parisse, Bernard Équilibre instable en régime semi-classique. I. Concentration microlocale, Commun. Partial Differ. Equations, Volume 19 (1994) no. 9-10, pp. 1535-1563 | Article | MR 1294470 | Zbl 0819.35116

[45] Vergini, Eduardo G.; Carlo, Gabriel G. Semiclassical quantization with short periodic orbits, J. Phys. A, Math. Gen., Volume 33 (2000) no. 25, pp. 4717-4724 | Article | MR 1777566 | Zbl 1004.81017

[46] Vergini, Eduardo G.; Carlo, Gabriel G. Semiclassical construction of resonances with hyperbolic structure: the scar function, J. Phys. A, Math. Gen., Volume 34 (2001) no. 21, pp. 4525-4552 | Article | MR 1835951 | Zbl 0978.81029

[47] Vergini, Eduardo G.; Schneider, David Asymptotic behaviour of matrix elements between scar functions, J. Phys. A, Math. Gen., Volume 38 (2005) no. 3, pp. 587-616 | Article | MR 2116626 | Zbl 1076.81019

[48] Voros, André Semi-classical approximations, Ann. Inst. Henri Poincaré Sect. A, Volume 24 (1976) no. 1, pp. 31-90 | MR 0420747

[49] Weinstein, Alan On Maslov’s quantization condition, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974) (Lecture Notes in Mathematics), Volume 459, Springer, 1975, pp. 341-372 | MR 0436231 | Zbl 0348.58016

[50] Zelditch, Steven Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987) no. 4, pp. 919-941 | Article | MR 916129 | Zbl 0643.58029

[51] Zelditch, Steven Wave invariants for non-degenerate closed geodesics, Geom. Funct. Anal., Volume 8 (1998) no. 1, pp. 179-217 | Article | MR 1601862 | Zbl 0908.58022

[52] Zelditch, Steven; Zworski, Maciej Ergodicity of eigenfunctions for ergodic billiards, Commun. Math. Phys., Volume 175 (1996) no. 3, pp. 673-682 http://projecteuclid.org/euclid.cmp/1104276097 | Article | MR 1372814 | Zbl 0840.58048

[53] Zworski, Maciej Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012, xii+431 pages | Article | MR 2952218 | Zbl 1252.58001