Resonance expansions of scattered waves

Tang, Siu-Hung; Zworski, Maciej

doi:10.1002/1097-0312(200010)53:10<1305::aid-cpa4>3.0.co;2-#

Cited by 73 publications

(61 citation statements)

References 29 publications

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“…These in turn allow a contour deformation argument leading to an expansion of waves in terms of quasinormal modes. Such expansions have a long tradition in scattering theory going back to Lax-Phillips and Vainberg-see [45] for the strongly trapping case and for references.…”

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confidence: 71%

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: 71%

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

Dyatlov

2012

Ann. Henri Poincaré

132

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“…Notamment pour des matrices de Toeplitz, intervenant lors de discrétisations d'opérateurs différentiels, le pseudospectre semble pouvoir jouer un role important dans l'étude de la stabilité du problème d'évolution discret correspondant ( [17]). Davies ([5]) propose une approche directe des problèmes d'évolution utilisant le pseudospectre, et les travaux antérieurs de Tang-Zworski ( [14]) et de Burq-Zworski ( [2]) illustrent bien les difficultés pseudospectrales pour les problèmes d'évolution. M.Zworski observe aussi que lors du calcul numérique des valeurs propres de certains opérateurs différentiels dans le cadre semiclassique, il semble y avoir un phénomène de migration des valeurs propres vers le bord du pseudospectre, dans la limite semiclassique( [20]).…”

Section: Introductionunclassified

Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints I : un modèle

Hager¹

2008

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Poles of the meromorphically continued Green's function are called resolvent resonances and physically represent "almost bound states." As with eigenvalues of the Laplacian, resolvent resonances determine the expansion in "normal modes" of solutions for the wave equation but unlike eigenvalues they represent "normal modes" whose energy in any bounded region decays with time (see, for example, [6], [33], and [37] for recent work on such "resonance wave expansions"). The energy decay is a consequence of the non-compactness of the underlying manifold.…”

Section: Resolvent Resonancesmentioning

confidence: 99%

Isoscattering deformations for complete manifolds of negative curvature

Perry

Schueth²

2006

J Geom Anal

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Abstract. We construct continuous families of non-isometric metrics on simply connected manifolds of dimension n ≥ 9 which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.

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Resonance expansions of scattered waves

Cited by 73 publications

References 29 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

Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints I : un modèle

Isoscattering deformations for complete manifolds of negative curvature

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