Strong scarring of logarithmic quasimodes
Annales de l'Institut Fourier, Volume 67 (2017) no. 6, p. 2307-2347

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 εℏ/|logℏ|, 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.

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 εℏ/|logℏ|, 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.

Received : 2015-08-17
Accepted : 2017-02-06
Published online : 2017-12-14
DOI : https://doi.org/10.5802/aif.3137
Classification:  35-xx,  58Jxx,  37-xx
Keywords: semiclassical analysis, quasimode, QUE, strong scarring 
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     pages = {2307-2347},
     doi = {10.5802/aif.3137},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2017__67_6_2307_0}
}

Eswarathasan, Suresh; Nonnenmacher, Stéphane. Strong scarring of logarithmic quasimodes. Annales de l'Institut Fourier, Volume 67 (2017) no. 6, pp. 2307-2347. doi : 10.5802/aif.3137. https://aif.centre-mersenne.org/item/AIF_2017__67_6_2307_0/

[1] Anantharaman, Nalini Entropy and the localization of eigenfunctions, Ann. Math., Tome 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, Tome 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., Tome 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.) Tome 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, Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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, Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 71 (1993) no. 2, pp. 559-607 | Article | MR 1233448 | Zbl 0788.35103

[22] Guillemin, Victor Wave-trace invariants, Duke Math. J., Tome 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., Tome 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., Tome 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 ℏ→0 limit for coherent states, Commun. Math. Phys., Tome 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., Tome 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., Tome 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, Tome 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, Tome 59 (1999) no. 6, pp. 6609-6628 | Article | MR 1695455

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

[32] Lindenstrauss, Elon Invariant measures and arithmetic quantum unique ergodicity, Ann. Math., Tome 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., Tome 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., Tome 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., Tome 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., Tome 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., Tome 36 (2003) no. 2, pp. 93-113 | MR 2021528 | Zbl 1060.35096

[39] Šnirelʼman, Alexander I. Ergodic properties of eigenfunctions, Usp. Mat. Nauk, Tome 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., Tome 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., Tome 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., Tome 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., Tome 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, Tome 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., Tome 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., Tome 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., Tome 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, Tome 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) Tome 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., Tome 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., Tome 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., Tome 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, Tome 138, American Mathematical Society, 2012, xii+431 pages | Article | MR 2952218 | Zbl 1252.58001