Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

Request Permissions   Purchase Content 
 
 

 

Resonance projectors and asymptotics for $ r$-normally hyperbolic trapped sets


Author: Semyon Dyatlov
Journal: J. Amer. Math. Soc. 28 (2015), 311-381
MSC (2010): Primary 35B34; Secondary 35S30, 37D05, 83C57
DOI: https://doi.org/10.1090/S0894-0347-2014-00822-5
Published electronically: December 16, 2014
MathSciNet review: 3300697
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a Weyl law for scattering resonances in a strip near the real axis when the trapped set is $ r$-normally hyperbolic with $ r$ large and a pinching condition on the normal expansion rates holds. Our dynamical assumptions are stable under smooth perturbations and are motivated by wave dynamics for black holes. The key step is a construction of a Fourier integral operator which microlocally projects onto the resonant states. In addition to the Weyl law, this operator provides new information about microlocal properties of resonant states.


References [Enhancements On Off] (What's this?)

  • [AgCo] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269-279. MR 0345551 (49 #10287)
  • [BFRZ] Jean-François Bony, Setsuro Fujiié, Thierry Ramond, and Maher Zerzeri, Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1351-1406 (2012) (English, with English and French summaries). MR 2951496, https://doi.org/10.5802/aif.2643
  • [BuZw] Nicolas Burq and Maciej Zworski, Control for Schrödinger operators on tori, Math. Res. Lett. 19 (2012), no. 2, 309-324. MR 2955763, https://doi.org/10.4310/MRL.2012.v19.n2.a4
  • [DaDy] Kiril Datchev and Semyon Dyatlov, Fractal Weyl laws for asymptotically hyperbolic manifolds, Geom. Funct. Anal. 23 (2013), no. 4, 1145-1206. MR 3077910, https://doi.org/10.1007/s00039-013-0225-8
  • [DDZ] Kiril Datchev, Semyon Dyatlov, and Maciej Zworski, Sharp polynomial bounds on the number of Pollicott-Ruelle resonances, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1168-1183. MR 3227152, https://doi.org/10.1017/etds.2013.3
  • [DiSj] 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 (2001b:35237)
  • [Do] Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357-390. MR 1626749 (99g:58073), https://doi.org/10.2307/121012
  • [DuGu] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39-79. MR 0405514 (53 #9307)
  • [Dy11a] Semyon Dyatlov, Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole, Comm. Math. Phys. 306 (2011), no. 1, 119-163. MR 2819421 (2012g:58055), https://doi.org/10.1007/s00220-011-1286-x
  • [Dy11b] Semyon Dyatlov, Exponential energy decay for Kerr-de Sitter black holes beyond event horizons, Math. Res. Lett. 18 (2011), no. 5, 1023-1035. MR 2875874, https://doi.org/10.4310/MRL.2011.v18.n5.a19
  • [Dy12] Semyon Dyatlov, Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes, Ann. Henri Poincaré 13 (2012), no. 5, 1101-1166. MR 2935116, https://doi.org/10.1007/s00023-012-0159-y
  • [Dy13] Semyon Dyatlov, Asymptotics of linear waves and resonances with applications to black holes, to appear in Comm. Math. Phys., arXiv:1305.1723.
  • [DyGu] Semyon Dyatlov and Colin Guillarmou, Microlocal limits of plane waves and Eisenstein functions, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 2, 371-448 (English, with English and French summaries). MR 3215926
  • [FaSj] Frédéric Faure and Johannes Sjöstrand, Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys. 308 (2011), no. 2, 325-364 (English, with English and French summaries). MR 2851145, https://doi.org/10.1007/s00220-011-1349-z
  • [FaTs12] Frédéric Faure and Masato Tsujii, Prequantum transfer operator for Anosov diffeomorphism, preprint, arXiv:206.0282.
  • [FaTs13a] Frédéric Faure and Masato Tsujii, Band structure of the Ruelle spectrum of contact Anosov flows, C. R. Math. Acad. Sci. Paris 351 (2013), no. 9-10, 385-391 (English, with English and French summaries). MR 3072166, https://doi.org/10.1016/j.crma.2013.04.022
  • [FaTs13b] Frédéric Faure and Masato Tsujii, The semiclassical zeta function for geodesic flows on negatively curved manifolds, preprint, arXiv:1311.4932.
  • [GéSj87] C. Gérard and J. Sjöstrand, Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys. 108 (1987), no. 3, 391-421. MR 874901 (88k:58151)
  • [GéSj88] C. Gérard and J. Sjöstrand, Resonances en limite semiclassique et exposants de Lyapunov, Comm. Math. Phys. 116 (1988), no. 2, 193-213 (French, with English summary). MR 939046 (89f:35057)
  • [GoSi] I. C. Gohberg and E. I. Sigal, An operator generalization of the logarithmic residue theorem and Rouché's theorem, Mat. Sb. (N.S.) 84(126) (1971), 607-629 (Russian). MR 0313856 (47 #2409)
  • [GSWW] Arseni Goussev, Roman Schubert, Holger Waalkens, and Stephen Wiggins, Quantum theory of reactive scattering in phase space, Adv. Quant. Chem. 60(2010), 269-332.
  • [GrSj] Alain Grigis and Johannes Sjöstrand, Microlocal analysis for differential operators, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. An introduction. MR 1269107 (95d:35009)
  • [Gu] Colin Guillarmou, Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), no. 1, 1-37. MR 2153454 (2006k:58051), https://doi.org/10.1215/S0012-7094-04-12911-2
  • [GuSt90] Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965 (58 #24404)
  • [GuSt13] Victor Guillemin and Shlomo Sternberg, Semi-classical analysis, International Press, Boston, MA, 2013. MR 3157301
  • [GLZ] Laurent Guillopé, Kevin K. Lin, and Maciej Zworski, The Selberg zeta function for convex co-compact Schottky groups, Comm. Math. Phys. 245 (2004), no. 1, 149-176. MR 2036371 (2005f:11193), https://doi.org/10.1007/s00220-003-1007-1
  • [HPS] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173 (58 #18595)
  • [Hö] Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. MR 0609014 (58 #29418)
  • [HöI] Lars Hörmander, The analysis of linear partial differential operators. I, Distribution theory and Fourier analysis, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990. MR 1065993 (91m:35001a)
  • [HöIII] Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1994. Pseudo-differential operators; Corrected reprint of the 1985 original. MR 1313500 (95h:35255)
  • [HöIV] Lars Hörmander, The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1994. Fourier integral operators; Corrected reprint of the 1985 original. MR 1481433 (98f:35002)
  • [ISZ] A. Iantchenko, J. Sjöstrand, and M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett. 9 (2002), no. 2-3, 337-362. MR 1909649 (2003f:35284), https://doi.org/10.4310/MRL.2002.v9.n3.a9
  • [JaNa] Dmitry Jakobson and Frédéric Naud, Lower bounds for resonances of infinite-area Riemann surfaces, Anal. PDE 3 (2010), no. 2, 207-225. MR 2657455 (2011m:58053), https://doi.org/10.2140/apde.2010.3.207
  • [KoSc] Kostas D. Kokkotas and Bernd G. Schmidt, Quasi-normal modes of stars and black holes, Living Rev. Relativ. 2 (1999), 1999-2, 73 pp. (electronic). MR 1713080 (2000f:83020), https://doi.org/10.12942/lrr-1999-2
  • [Li] Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2) 159 (2004), no. 3, 1275-1312. MR 2113022 (2005k:37048), https://doi.org/10.4007/annals.2004.159.1275
  • [Ma] A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, vol. 71, American Mathematical Society, Providence, RI, 1988. Translated from the Russian by H. H. McFaden; Translation edited by Ben Silver; With an appendix by M. V. Keldysh. MR 971506 (89h:47023)
  • [MaMe] Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260-310. MR 916753 (89c:58133), https://doi.org/10.1016/0022-1236(87)90097-8
  • [MeZw] Richard Melrose and Maciej Zworski, Scattering metrics and geodesic flow at infinity, Invent. Math. 124 (1996), no. 1-3, 389-436. MR 1369423 (96k:58230), https://doi.org/10.1007/s002220050058
  • [Mü] Werner Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math. 109 (1992), no. 2, 265-305. MR 1172692 (93g:58151), https://doi.org/10.1007/BF01232028
  • [No] Stéphane Nonnenmacher, Spectral problems in open quantum chaos, Nonlinearity 24(2011), R123-R167.
  • [NSZ11] Stéphane Nonnenmacher, Johannes Sjöstrand, and Maciej Zworski, From open quantum systems to open quantum maps, Comm. Math. Phys. 304 (2011), no. 1, 1-48. MR 2793928 (2012m:58018), https://doi.org/10.1007/s00220-011-1214-0
  • [NSZ14] Stéphane Nonnenmacher, Johannes Sjöstrand, and Maciej Zworski, Fractal Weyl law for open quantum chaotic maps, Ann. of Math. (2) 179 (2014), no. 1, 179-251. MR 3126568, https://doi.org/10.4007/annals.2014.179.1.3
  • [NoZw09] Stéphane Nonnenmacher and Maciej Zworski, Quantum decay rates in chaotic scattering, Acta Math. 203 (2009), no. 2, 149-233. MR 2570070 (2011c:58063), https://doi.org/10.1007/s11511-009-0041-z
  • [NoZw13] Stéphane Nonnenmacher and Maciej Zworski, Decay of correlations for normally hyperbolic trapping, to appear in Invent. Math., arXiv:1302.4483.
  • [Re] T. Regge, Analytic properties of the scattering matrix, Nuovo Cimento (10) 8 (1958), 671-679. MR 0095702 (20 #2203)
  • [SáZw] Antônio Sá Barreto and Maciej Zworski, Distribution of resonances for spherical black holes, Math. Res. Lett. 4 (1997), no. 1, 103-121. MR 1432814 (97m:83063), https://doi.org/10.4310/MRL.1997.v4.n1.a10
  • [Sj90] Johannes Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1-57. MR 1047116 (91e:35166), https://doi.org/10.1215/S0012-7094-90-06001-6
  • [Sj97] J. Sjöstrand, A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory (Lucca, 1996) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 490, Kluwer Acad. Publ., Dordrecht, 1997, pp. 377-437. MR 1451399 (99e:47064)
  • [Sj00] Johannes Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36 (2000), no. 5, 573-611 (English, with English and French summaries). MR 1798488 (2001k:35233), https://doi.org/10.2977/prims/1195142811
  • [Sj01] Johannes Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr. 221 (2001), 95-149. MR 1806367 (2001k:58063), https://doi.org/10.1002/1522-2616(200101)221:1$ \langle $95::AID-MANA95$ \rangle $3.0.CO;2-P
  • [Sj11] Johannes Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials, preprint, arXiv:1111.3549.
  • [SjVo] Johannes Sjöstrand and Georgi Vodev, Asymptotics of the number of Rayleigh resonances, Math. Ann. 309 (1997), no. 2, 287-306. With an appendix by Jean Lannes. MR 1474193 (99g:35097), https://doi.org/10.1007/s002080050113
  • [SjZw91] Johannes Sjöstrand and Maciej Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (1991), no. 4, 729-769. MR 1115789 (92g:35166), https://doi.org/10.2307/2939287
  • [SjZw99] Johannes Sjöstrand and Maciej Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183 (1999), no. 2, 191-253. MR 1738044 (2001g:35204), https://doi.org/10.1007/BF02392828
  • [SjZw07] Johannes Sjöstrand and Maciej Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J. 137 (2007), no. 3, 381-459. MR 2309150 (2008e:35037), https://doi.org/10.1215/S0012-7094-07-13731-1
  • [StVo] P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body, Duke Math. J. 78 (1995), no. 3, 677-714. MR 1334206 (96e:35034), https://doi.org/10.1215/S0012-7094-95-07825-9
  • [TaZw] Siu-Hung Tang and Maciej Zworski, From quasimodes to reasonances, Math. Res. Lett. 5 (1998), no. 3, 261-272. MR 1637824 (99i:47088), https://doi.org/10.4310/MRL.1998.v5.n3.a1
  • [Ta] Michael E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR 1395148 (98b:35002b)
  • [Ti] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550 (88c:11049)
  • [Ts] Masato Tsujii, Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform, Ergodic Theory Dynam. Systems 32 (2012), no. 6, 2083-2118. MR 2995886, https://doi.org/10.1017/S0143385711000605
  • [Va12] András Vasy, Microlocal analysis of asymptotically hyperbolic spaces and high-energy resolvent estimates, Inverse problems and applications: inside out. II, Math. Sci. Res. Inst. Publ., vol. 60, Cambridge Univ. Press, Cambridge, 2013, pp. 487-528. MR 3135765
  • [Va13] András Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math. 194 (2013), no. 2, 381-513. MR 3117526, https://doi.org/10.1007/s00222-012-0446-8
  • [VũNg] San Vũ Ngoc, Systèmes intégrables semi-classiques: du local au global, Panoramas et Synthèses [Panoramas and Syntheses], vol. 22, Société Mathématique de France, Paris, 2006 (French, with English and French summaries). MR 2331010 (2008e:37056)
  • [WuZw11] Jared Wunsch and Maciej Zworski, Resolvent estimates for normally hyperbolic trapped sets, Ann. Henri Poincaré 12 (2011), no. 7, 1349-1385. MR 2846671, https://doi.org/10.1007/s00023-011-0108-1
  • [Zw87] Maciej Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), no. 2, 277-296. MR 899652 (88h:81223), https://doi.org/10.1016/0022-1236(87)90069-3
  • [Zw] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 35B34, 35S30, 37D05, 83C57

Retrieve articles in all journals with MSC (2010): 35B34, 35S30, 37D05, 83C57


Additional Information

Semyon Dyatlov
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: dyatlov@math.mit.edu

DOI: https://doi.org/10.1090/S0894-0347-2014-00822-5
Received by editor(s): February 21, 2013
Published electronically: December 16, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society