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

Abstract

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], ...

Pollicott-Ruelle Resonant States and Betti Numbers

“…Such correspondences have recently been developed in various contexts (see [DFG15] for compact hyperbolic manifolds, [GHW18,Had18] for the convex co-compact setting, and [GHW20] for generalizations to general rank one manifolds). We will use the general framework for vector bundles developed by the authors in [KW19]. Additionally we use a Poisson transform due to Gaillard [Gai86] and combining both ingredients allows us to construct an explicit bijection between the Pollicott-Ruelle resonant states in perpendicular one forms and the kernel of the Hodge Laplacian.…”

“…There exists a very efficient Lie-theoretic language to describe the structure of M, the co-sphere bundle S * M, as well as the invariant vector bundles which we introduce in this subsection. For more details we refer the reader to [GHW20,KW19] and for background information to the textbooks [Kna02,Hel01]. In the following we shall introduce the required abstract language in a quite concrete way, tailored to the particular group G = SO(n + 1, 1) 0 .…”

“…Furthermore, the resonance zero has no Jordan block and if n ≥ 3, then zero is the unique leading resonance and there is a spectral gap. 3 We prove these statements using the general framework of vector-valued quantumclassical correspondence developed by the authors [KW19] as well as a Poisson transform of Gaillard [Gai86]. 4 Without any further effort these ingredients provide additional examples of resonance multiplicities related to not only the first but to all Betti numbers, see Proposition 2.3.…”

“…(3) By [KW19] there are no first band Jordan blocks, so it follows that u is actually a resonant state. (4) Since u is a first band resonant state, u corresponds to a distributional one-form u ∞ on the sphere S n , the boundary at infinity of the hyperbolic space H n+1 .…”

“…Open Access funding provided by Projekt DEAL. After the appearance of [KW19], Semyon Dyatlov raised the question whether the vector-valued quantum-classical correspondence might shed light on generalizations of [DZ17] to higher dimensions. We are grateful to him for proposing this question as well as for several helpful discussions.…”