Power spectrum of the geodesic flow on hyperbolic manifolds

Dyatlov, Semyon; Faure, Frédéric; Guillarmou, Colin

doi:10.2140/apde.2015.8.923

Cited by 49 publications

(132 citation statements)

References 53 publications

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“…In particular, we show in Appendix B how to determine the joint eigenspaces on some particular associated vector bundles over H 3 = SO(3, 1) 0 /SO(3). This allows us to recover the results obtained in [DFG15] and even give a slightly more precise description.…”

Section: Introductionsupporting

confidence: 63%

“…Remark 1. An exact band structure has before been obtained in [DFG15] in the scalar case for manifolds of constant negative curvature. We would like to emphasize that even the step to generalize the band structure for scalar resonances from constant curvature manifolds to general rank one manifolds heavily relies on the quantum-classical correspondence on vector bundles given in Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces

Küster

Weich

2019

International Mathematics Research Notices

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For a compact Riemannian locally symmetric space M of rank one and an associated vector bundle V τ over the unit cosphere bundle S * M, we give a precise description of those classical (Pollicott-Ruelle) resonant states on V τ that vanish under covariant derivatives in the Anosovunstable directions of the chaotic geodesic flow on S * M. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators D(G, σ) on compatible associated vector bundles W σ over M. As a consequence of this description, we obtain an exact band structure of the Pollicott-Ruelle spectrum. Further, under some mild assumptions on the representations τ and σ defining the bundles V τ and W σ , we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott-Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of W σ . Our methods of proof are based on representation theory and Lie theory.and call the elements of Res 0 (λ) first band resonant states. We call λ a first band resonance 2 if Res 0 (λ) = {0}. Now, let π : S * M → M be the bundle projection and recall that the pushforward of distributions along this submersion is a well-defined continuous operator π * : D ′ (S * M) → D ′ (M). Dyatlov, Faure, and Guillarmou then prove the following remarkable result [DFG15]: If λ is a first band Pollicott-Ruelle resonance, π * restricts towhere µ(λ) = − (λ + ̺) 2 +̺ 2 and ̺ = (dim(M)−1)/2. Furthermore, provided one has λ / ∈ −̺− 1 2 N 0 , the above map is an isomorphism of complex vector spaces.The aim of the present article is to extend these results to the setting of associated vector bundles over arbitrary compact Riemannian locally symmetric spaces of rank one. From now on, suppose that M is such a space. Let us briefly introduce the setting: We can write our manifold as a double quotient M = Γ\G/K, where G is a connected real semisimple Lie group of real rank one with finite center, K ⊂ G a maximally compact subgroup, and Γ ⊂ G a discrete torsion-free cocompact subgroup. The Riemannian metric on M is defined in a canonical way from the non-degenerate Killing form on the Lie algebra g of G. This metric has strictly negative sectional curvature 3 , so 1 In the language of physics, the geodesic flow ϕt : S * M → S * M generated by X describes the classical mechanical movement of a single free point particle on M with constant speed, while ∆ is, up to rescaling, the Schrödinger operator associated to the corresponding quantum mechanical system. 2 The name "first band" comes from the observation due to Dyatlov, Faure, and Guillarmou that the Pollicott-Ruelle spectrum on surfaces of strictly negative constant curvature has an exact structure of bands parallel to the imaginary axis. The first band resonances turn out to be those lying closest to the imaginary axis.3 The case of manifolds with strictly negative constant curvature, studied in [DFG15], ...

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Section: Introductionsupporting

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces

Küster

Weich

2019

International Mathematics Research Notices

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“…The zeros and poles of the Ruelle zeta function are related to the Pollicott-Ruelle resonances, while the singularities of the Selberg zeta function correspond to quasinormal modes and thus eigenvalues of the Laplacian (see [35] and references therein). The Selberg trace formula and zeta function also appear in the literature on chaos [15], and so perhaps the results presented here will have further applications in that subfield.…”

Section: Discussionmentioning

confidence: 99%

Connecting quasinormal modes and heat kernels in 1-loop determinants

2020

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We connect two different approaches for calculating functional determinants on quotients of hyperbolic spacetime: the heat kernel method and the quasinormal mode method. For the example of a rotating BTZ background, we show how the image sum in the heat kernel method builds up the logarithms in the quasinormal mode method, while the thermal sum in the quasinormal mode method builds up the integrand of the heat kernel. More formally, we demonstrate how the heat kernel and quasinormal mode methods are linked via the Selberg zeta function. We show that a 1-loop partition function computed using the heat kernel method may be cast as a Selberg zeta function whose zeros encode quasinormal modes. We discuss how our work may be used to predict quasinormal modes on more complicated spacetimes. arXiv:1811.08433v2 [hep-th]

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“…[71]. Although these resonances have been known for a long time in the case of surfaces, in higher dimensions some new structure was discovered in [71], see also Guillarmou-Hilgert-Weich [118].…”

Section: Distribution Of Pollicott-ruelle Resonancesmentioning

confidence: 98%