Resonance projectors and asymptotics for 𝑟-normally hyperbolic trapped sets

Dyatlov, Semyon

doi:10.1090/S0894-0347-2014-00822-5

Resonance projectors and asymptotics for $r$-normally hyperbolic trapped sets
HTML articles powered by AMS MathViewer

by Semyon Dyatlov;

J. Amer. Math. Soc. 28 (2015), 311-381

DOI: https://doi.org/10.1090/S0894-0347-2014-00822-5

Published electronically: December 16, 2014

PDF | Request permission

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

J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269–279. MR 345551, DOI 10.1007/BF01877510
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, DOI 10.5802/aif.2643
Nicolas Burq and Maciej Zworski, Control for Schrödinger operators on tori, Math. Res. Lett. 19 (2012), no. 2, 309–324. MR 2955763, DOI 10.4310/MRL.2012.v19.n2.a4
Kiril Datchev and Semyon Dyatlov, Fractal Weyl laws for asymptotically hyperbolic manifolds, Geom. Funct. Anal. 23 (2013), no. 4, 1145–1206. MR 3077910, DOI 10.1007/s00039-013-0225-8
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, DOI 10.1017/etds.2013.3
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
Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357–390. MR 1626749, DOI 10.2307/121012
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 405514, DOI 10.1007/BF01405172
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, DOI 10.1007/s00220-011-1286-x
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, DOI 10.4310/MRL.2011.v18.n5.a19
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, DOI 10.1007/s00023-012-0159-y
Semyon Dyatlov, Asymptotics of linear waves and resonances with applications to black holes, to appear in Comm. Math. Phys., arXiv:1305.1723.
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, DOI 10.24033/asens.2217
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, DOI 10.1007/s00220-011-1349-z
Frédéric Faure and Masato Tsujii, Prequantum transfer operator for Anosov diffeomorphism, preprint, arXiv:206.0282.
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, DOI 10.1016/j.crma.2013.04.022
Frédéric Faure and Masato Tsujii, The semiclassical zeta function for geodesic flows on negatively curved manifolds, preprint, arXiv:1311.4932.
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, DOI 10.1007/BF01212317
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, DOI 10.1007/BF01225255
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 313856
Arseni Goussev, Roman Schubert, Holger Waalkens, and Stephen Wiggins, Quantum theory of reactive scattering in phase space, Adv. Quant. Chem. 60(2010), 269–332.
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, DOI 10.1017/CBO9780511721441
Colin Guillarmou, Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), no. 1, 1–37. MR 2153454, DOI 10.1215/S0012-7094-04-12911-2
Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, RI, 1977. MR 516965, DOI 10.1090/surv/014
Victor Guillemin and Shlomo Sternberg, Semi-classical analysis, International Press, Boston, MA, 2013. MR 3157301
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, DOI 10.1007/s00220-003-1007-1
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 501173, DOI 10.1007/BFb0092042
Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 609014, DOI 10.1007/BF02391913
Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065993, DOI 10.1007/978-3-642-61497-2
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
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
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, DOI 10.4310/MRL.2002.v9.n3.a9
Dmitry Jakobson and Frédéric Naud, Lower bounds for resonances of infinite-area Riemann surfaces, Anal. PDE 3 (2010), no. 2, 207–225. MR 2657455, DOI 10.2140/apde.2010.3.207
Kostas D. Kokkotas and Bernd G. Schmidt, Quasi-normal modes of stars and black holes, Living Rev. Relativ. 2 (1999), 1999-2, 73. MR 1713080, DOI 10.12942/lrr-1999-2
Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2) 159 (2004), no. 3, 1275–1312. MR 2113022, DOI 10.4007/annals.2004.159.1275
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, DOI 10.1090/mmono/071
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, DOI 10.1016/0022-1236(87)90097-8
Richard Melrose and Maciej Zworski, Scattering metrics and geodesic flow at infinity, Invent. Math. 124 (1996), no. 1-3, 389–436. MR 1369423, DOI 10.1007/s002220050058
Werner Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math. 109 (1992), no. 2, 265–305. MR 1172692, DOI 10.1007/BF01232028
Stéphane Nonnenmacher, Spectral problems in open quantum chaos, Nonlinearity 24(2011), R123–R167.
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, DOI 10.1007/s00220-011-1214-0
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, DOI 10.4007/annals.2014.179.1.3
Stéphane Nonnenmacher and Maciej Zworski, Quantum decay rates in chaotic scattering, Acta Math. 203 (2009), no. 2, 149–233. MR 2570070, DOI 10.1007/s11511-009-0041-z
Stéphane Nonnenmacher and Maciej Zworski, Decay of correlations for normally hyperbolic trapping, to appear in Invent. Math., arXiv:1302.4483.
T. Regge, Analytic properties of the scattering matrix, Nuovo Cimento (10) 8 (1958), 671–679. MR 95702, DOI 10.1007/BF02815247
Antônio Sá Barreto and Maciej Zworski, Distribution of resonances for spherical black holes, Math. Res. Lett. 4 (1997), no. 1, 103–121. MR 1432814, DOI 10.4310/MRL.1997.v4.n1.a10
Johannes Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1–57. MR 1047116, DOI 10.1215/S0012-7094-90-06001-6
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
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, DOI 10.2977/prims/1195142811
Johannes Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr. 221 (2001), 95–149. MR 1806367, DOI 10.1002/1522-2616(200101)221:1<95::AID-MANA95>3.0.CO;2-P
Johannes Sjöstrand, Weyl law for semi-classical resonances with randomly perturbed potentials, preprint, arXiv:1111.3549.
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, DOI 10.1007/s002080050113
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, DOI 10.1090/S0894-0347-1991-1115789-9
Johannes Sjöstrand and Maciej Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183 (1999), no. 2, 191–253. MR 1738044, DOI 10.1007/BF02392828
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, DOI 10.1215/S0012-7094-07-13731-1
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, DOI 10.1215/S0012-7094-95-07825-9
Siu-Hung Tang and Maciej Zworski, From quasimodes to resonances, Math. Res. Lett. 5 (1998), no. 3, 261–272. MR 1637824, DOI 10.4310/MRL.1998.v5.n3.a1
Michael E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR 1395148, DOI 10.1007/978-1-4684-9320-7
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
Masato Tsujii, Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform, Ergodic Theory Dynam. Systems 32 (2012), no. 6, 2083–2118. MR 2995886, DOI 10.1017/S0143385711000605
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
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, DOI 10.1007/s00222-012-0446-8
San Vũ Ngọc, 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
Jared Wunsch and Maciej Zworski, Resolvent estimates for normally hyperbolic trapped sets, Ann. Henri Poincaré 12 (2011), no. 7, 1349–1385. MR 2846671, DOI 10.1007/s00023-011-0108-1
Maciej Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), no. 2, 277–296. MR 899652, DOI 10.1016/0022-1236(87)90069-3
Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218, DOI 10.1090/gsm/138