Abstract
The usual semi-classical correspondence (called quantum-classical) shows that the fixed-time evolution of wave packets by a wave equation reveals the geodesic flow in the small-wavelength limit λ → 0. This geodesic flow is determined by the principal symbol of the wave operator. So different operators with different spectra can have the same classical limit. The Duistermaat-Guillemin trace formula shows that the spectrum of the operator determines the set of periodic geodesic lengths, but not vice versa.
We wish to show the opposite direction : the geodesic flow when it is Anosov, determines a unique wave equation generated by an operator equivalent to √∆ at leading order and whose spectrum is characterized by the periodic geodesics, via a zeta function.
This wave equation appears dynamically as follows. In the simple case of a hyperbolic surface N (i.e. smooth, compact of curvature -1), the spherical mean at time t ∈ R of a function u0 : N → C is the function ut where at each point x ∈ N , the value ut(x) is the mean of u0 over the geodesic circle of center x and radius |t|. For t → ∞, each circle becomes dense and ut converges exponentially fast to the spatial mean ⟨u0⟩of u0. We are interested in the fluctuations around this mean by positing vt = e|t|/2 (ut - ⟨u0⟩). The surprise is that these fluctuations are solution of the wave equation on N. It will be shown that such a phenomenon is more general to any Anosov Riemannian variety giving an emergent wave equation, generated by an operator that is a kind of "dynamical quantization" of the classical flow.
The ideas and ingredients that enable these results to be obtained will be presented, and include microlocal analysis, anisotropic Sobolev spaces, Ruelle spectra and symplectic spineurs.
Work in collaboration with Masato Tsujii, arxiv 2102.11196.
