Resolvent Estimates for Normally Hyperbolic Trapped Sets

Wunsch, Jared; Zworski, Maciej

doi:10.1007/s00023-011-0108-1

Cited by 76 publications

(178 citation statements)

References 50 publications

(72 reference statements)

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“…Vasy [50] has recently obtained a microlocal description of the scattering resolvent and in particular recovered the results of [26,27] on meromorphy of the resolvent and exponential decay; see [50,Appendix] for how his work relates to [26]. The crucial component for obtaining exponential decay was the work of WunschZworski [52] on resolvent estimates for normally hyperbolic trapping.…”

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confidence: 91%

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

Dyatlov

2012

Ann. Henri Poincaré

132

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Abstract. We establish a Bohr-Sommerfeld type condition for quasinormal modes of a slowly rotating Kerr-de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a = 0 (Schwarzschild-de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.Quasi-normal modes (QNMs) of black holes are a topic of continued interest in theoretical physics: from the classical interpretation as ringdown of gravitational waves [11] to the recent investigations in the context of string theory [33]. The ringdown 1 plays a role in experimental projects aimed at the detection of gravitational waves, such as LIGO [1]. See [35] for an overview of the vast physics literature on the topic and [6,36,37,53] for some more recent developments.In this paper we consider the Kerr-de Sitter model of a rotating black hole and assume that the speed of rotation a is small; for a = 0, one gets the stationary Schwarzschild-de Sitter black hole. The de Sitter model corresponds to assuming that the cosmological constant Λ is positive, which is consistent with the current Lambda-CDM standard model of cosmology.

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confidence: 91%

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

Dyatlov

2012

Ann. Henri Poincaré

132

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“…We remark that, in the setting of [21,30], (1.2) holds with C 0 h −k replaced by C 0 (log h −1 )h −1 , and so the improvement in our result is only of a factor of log(1/h). On the other hand, in [1], Bony, Burq and Ramond prove that for P a semiclassical Schrödinger operator on R n , the presence of a single trapped trajectory implies that…”

Section: Annales De L'institut Fouriermentioning

confidence: 73%

“…[23] for an example and [1] for a recent introduction to the subject of semiclassical resolvent estimates). Nonetheless, (1.2) is satisfied for many hyperbolic trapped geometries, including those studied in [21,30]. See [12,Theorem 6.1] for (1.2) in the asymptotically hyperbolic case, and see [11] and [30,Corollary 1] for the asymptotically conic case.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…Nonetheless, (1.2) is satisfied for many hyperbolic trapped geometries, including those studied in [21,30]. See [12,Theorem 6.1] for (1.2) in the asymptotically hyperbolic case, and see [11] and [30,Corollary 1] for the asymptotically conic case. Bony and Petkov [2] prove (1.2) for a general "black box" perturbation of the Laplacian in R n assuming only that there is a resonance-free strip, and it is likely that this condition suffices for asymptotically conic or hyperbolic manifolds as well.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…See [21] and [30] for examples of theorems about resolvent estimates in the presence of trapping which are simplified in this setting, and see [12] for a general method for gluing in another (more interesting) semiclassically outgoing infinity once such an estimate is proved. This method is used in the present paper to construct the example in §5.…”

Section: Complex Absorbing Barriersmentioning

confidence: 99%

See 2 more Smart Citations

Propagation through trapped sets and semiclassical resolvent estimates

Datchev¹,

Vasy²

2012

Ann. inst. Fourier

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Propagation Through Trapped Sets and Semiclassical Resolvent Estimates

Datchev

Vasy

2012

Microlocal Methods in Mathematical Physics and Global Analysis

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Abstract. Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.

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Resolvent Estimates for Normally Hyperbolic Trapped Sets

Cited by 76 publications

References 50 publications

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

Propagation through trapped sets and semiclassical resolvent estimates

Propagation Through Trapped Sets and Semiclassical Resolvent Estimates

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