Spectral gaps for normally hyperbolic trapping
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, p. 55-82

We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming/outgoing tails. An important application is wave decay on Kerr and Kerr–de Sitter black holes. We recover the optimal size of the strip and give an o(h -2 ) resolvent bound there. We next show existence of deeper resonance free strips under the r-normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the one-sided cutoff resolvent on the real line.

Cet article démontre l’existence d’une bande sans résonances pour des ensembles captés normalement hyperboliques de codimension 2, dont les variétés entrantes/sortantes sont lisses. Une application importante est la décroissance exponentielle des ondes pour les trous noirs de Kerr et Kerr–de Sitter. On retrouve la taille optimale de la bande et on y donne une borne o(h -2 ) de la résolvante. On démontre alors l’existence de bandes plus profondes sans résonances si l’ensemble capté est r-normalement hyperbolique et satisfait une condition de pincement. On donne aussi une borne inférieure sur la norme de la résolvante tronquée sur l’axe réel.

Received : 2014-04-15
Revised : 2015-03-06
Accepted : 2015-03-26
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3005
Classification:  35B34,  37D05
Keywords: spectral gaps, normally hyperbolic trapping, black holes
@article{AIF_2016__66_1_55_0,
     author = {Dyatlov, Semyon},
     title = {Spectral gaps for normally hyperbolic trapping},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {55-82},
     doi = {10.5802/aif.3005},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_1_55_0}
}
Dyatlov, Semyon. Spectral gaps for normally hyperbolic trapping. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 55-82. doi : 10.5802/aif.3005. http://www.numdam.org/item/AIF_2016__66_1_55_0/

[1] Burq, Nicolas Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math., Tome 180 (1998) no. 1, pp. 1-29 | Article

[2] Burq, Nicolas Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math., Tome 124 (2002) no. 4, pp. 677-735

[3] Burq, Nicolas; Zworski, Maciej Geometric control in the presence of a black box, J. Amer. Math. Soc., Tome 17 (2004) no. 2, p. 443-471 (electronic) | Article

[4] Christianson, Hans Dispersive estimates for manifolds with one trapped orbit, Comm. Partial Differential Equations, Tome 33 (2008) no. 7-9, pp. 1147-1174 | Article

[5] Datchev, Kiril Quantitative limiting absorption principle in the semiclassical limit, Geom. Funct. Anal., Tome 24 (2014) no. 3, pp. 740-747 | Article

[6] Datchev, Kiril; Vasy, András Gluing semiclassical resolvent estimates via propagation of singularities, Int. Math. Res. Not. IMRN (2012) no. 23, pp. 5409-5443

[7] Datchev, Kiril; Vasy, András Propagation through trapped sets and semiclassical resolvent estimates, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 6, p. 2347-2377 (2013) | Article

[8] Datchev, Kiril; Vasy, András Semiclassical resolvent estimates at trapped sets, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 6, p. 2379-2384 (2013) | Article

[9] Dyatlov, Semyon Exponential energy decay for Kerr–de Sitter black holes beyond event horizons, Math. Res. Lett., Tome 18 (2011) no. 5, pp. 1023-1035 | Article

[10] Dyatlov, Semyon Asymptotics of linear waves and resonances with applications to black holes, Comm. Math. Phys., Tome 335 (2015) no. 3, pp. 1445-1485 | Article

[11] Dyatlov, Semyon Resonance projectors and asymptotics for r-normally hyperbolic trapped sets, J. Amer. Math. Soc., Tome 28 (2015) no. 2, pp. 311-381 | Article

[12] Dyatlov, Semyon; Faure, Frédéric; Guillarmou, Colin Power spectrum of the geodesic flow on hyperbolic manifolds (to appear in Analysis and PDE, http://arxiv.org/abs/1403.0256)

[13] Dyatlov, Semyon; Zworski, Maciej Dynamical zeta functions for Anosov flows via microlocal analysis (to appear in Annales de l’ENS, http://arxiv.org/abs/1306.4203)

[14] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances (http://math.mit.edu/~dyatlov/res/ )

[15] Faure, Fréderic; Tsujii, Masato The semiclassical zeta function for geodesic flows on negatively curved manifolds (http://arxiv.org/abs/1311.4932 )

[16] Gérard, C.; Sjöstrand, J. Resonances en limite semiclassique et exposants de Lyapunov, Comm. Math. Phys., Tome 116 (1988) no. 2, pp. 193-213 http://projecteuclid.org/euclid.cmp/1104161300

[17] Goussev, Arseni; Schubert, Robert; Holger, Weelkens; Stephen, Wiggins Quantum theory of reactive scattering in phase space, Adv. Quant. Chem., Tome 60 (2010), pp. 269-332

[18] Grigis, Alain; Sjöstrand, Johannes Microlocal analysis for differential operators, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 196 (1994), iv+151 pages (An introduction) | Article

[19] Hintz, Peter; Vasy, András Global well-posedness of quasilinear wave equations on asymptotically de Sitter spaces (http://arxiv.org/abs/1311.6859 )

[20] Hintz, Peter; Vasy, András Non-trapping estimates near normally hyperbolic trapping (http://arxiv.org/abs/1311.7197 )

[21] Hintz, Peter; Vasy, András Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes (http://arxiv.org/abs/1306.4705 )

[22] Hirsch, M. W.; Pugh, C. C.; Shub, M. Invariant manifolds, Springer-Verlag, Berlin-New York, Lecture Notes in Mathematics, Vol. 583 (1977), ii+149 pages

[23] Nonnenmacher, Stéphane; Zworski, Maciej Decay of correlations for normally hyperbolic trapping (http://arxiv.org/abs/1302.4483 )

[24] Royer, Julien Limiting absorption principle for the dissipative Helmholtz equation, Comm. Partial Differential Equations, Tome 35 (2010) no. 8, pp. 1458-1489 | Article

[25] Vasy, András Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math., Tome 194 (2013) no. 2, pp. 381-513 | Article

[26] Vodev, Georgi Exponential bounds of the resolvent for a class of noncompactly supported perturbations of the Laplacian, Math. Res. Lett., Tome 7 (2000) no. 2-3, pp. 287-298 | Article

[27] Wunsch, Jared Resolvent estimates with mild trapping (http://arxiv.org/abs/1209.0843 )

[28] Wunsch, Jared; Zworski, Maciej Resolvent estimates for normally hyperbolic trapped sets, Ann. Henri Poincaré, Tome 12 (2011) no. 7, pp. 1349-1385 | Article

[29] Zworski, Maciej Semiclassical analysis, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 138 (2012), xii+431 pages