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 resolvent bound there. We next show existence of deeper resonance free strips under the -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 de la résolvante. On démontre alors l’existence de bandes plus profondes sans résonances si l’ensemble capté est -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.
Revised:
Accepted:
Published online:
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}, pages = {55--82}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {1}, year = {2016}, doi = {10.5802/aif.3005}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3005/} }
TY - JOUR TI - Spectral gaps for normally hyperbolic trapping JO - Annales de l'Institut Fourier PY - 2016 DA - 2016/// SP - 55 EP - 82 VL - 66 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3005/ UR - https://doi.org/10.5802/aif.3005 DO - 10.5802/aif.3005 LA - en ID - AIF_2016__66_1_55_0 ER -
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3005/
[1] 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] Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math., Tome 124 (2002) no. 4, pp. 677-735
[3] Geometric control in the presence of a black box, J. Amer. Math. Soc., Tome 17 (2004) no. 2, p. 443-471 (electronic) | Article
[4] Dispersive estimates for manifolds with one trapped orbit, Comm. Partial Differential Equations, Tome 33 (2008) no. 7-9, pp. 1147-1174 | Article
[5] Quantitative limiting absorption principle in the semiclassical limit, Geom. Funct. Anal., Tome 24 (2014) no. 3, pp. 740-747 | Article
[6] Gluing semiclassical resolvent estimates via propagation of singularities, Int. Math. Res. Not. IMRN (2012) no. 23, pp. 5409-5443
[7] Propagation through trapped sets and semiclassical resolvent estimates, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 6, p. 2347-2377 (2013) | Article
[8] Semiclassical resolvent estimates at trapped sets, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 6, p. 2379-2384 (2013) | Article
[9] 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] Asymptotics of linear waves and resonances with applications to black holes, Comm. Math. Phys., Tome 335 (2015) no. 3, pp. 1445-1485 | Article
[11] Resonance projectors and asymptotics for -normally hyperbolic trapped sets, J. Amer. Math. Soc., Tome 28 (2015) no. 2, pp. 311-381 | Article
[12] Power spectrum of the geodesic flow on hyperbolic manifolds (to appear in Analysis and PDE, http://arxiv.org/abs/1403.0256)
[13] Dynamical zeta functions for Anosov flows via microlocal analysis (to appear in Annales de l’ENS, http://arxiv.org/abs/1306.4203)
[14] Mathematical theory of scattering resonances (http://math.mit.edu/~dyatlov/res/)
[15] The semiclassical zeta function for geodesic flows on negatively curved manifolds (http://arxiv.org/abs/1311.4932)
[16] 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] Quantum theory of reactive scattering in phase space, Adv. Quant. Chem., Tome 60 (2010), pp. 269-332
[18] Microlocal analysis for differential operators, London Mathematical Society Lecture Note Series, Tome 196, Cambridge University Press, Cambridge, 1994, iv+151 pages (An introduction) | Article
[19] Global well-posedness of quasilinear wave equations on asymptotically de Sitter spaces (http://arxiv.org/abs/1311.6859)
[20] Non-trapping estimates near normally hyperbolic trapping (http://arxiv.org/abs/1311.7197)
[21] Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes (http://arxiv.org/abs/1306.4705)
[22] Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977, ii+149 pages
[23] Decay of correlations for normally hyperbolic trapping (http://arxiv.org/abs/1302.4483)
[24] Limiting absorption principle for the dissipative Helmholtz equation, Comm. Partial Differential Equations, Tome 35 (2010) no. 8, pp. 1458-1489 | Article
[25] 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] 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] Resolvent estimates with mild trapping (http://arxiv.org/abs/1209.0843)
[28] Resolvent estimates for normally hyperbolic trapped sets, Ann. Henri Poincaré, Tome 12 (2011) no. 7, pp. 1349-1385 | Article
[29] Semiclassical analysis, Graduate Studies in Mathematics, Tome 138, American Mathematical Society, Providence, RI, 2012, xii+431 pages
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