From geodesic flow to wave dynamics on hyperbolic surfaces

We study the geodesic flow on the unit cotangent bundle $M=S^{*}\mathcal{N}$ of a closed hyperbolic surface $\mathcal{N}$, using the representation theory of $SL_{2}(\mathbb{R})$. We construct explicit $X$-adapted Hilbert spaces, obtained by completing propagated dense domains of $L^{2}(M)$, which are tailored to the spectral analysis of the geodesic generator $X$. In these spaces, $X$ becomes a normal operator with discrete spectrum, except at the threshold $\mu=1/4$, where Jordan blocks of size two may occur. In this Hilbert model, the propagator $e^{tX}$ factorizes into a damped harmonic oscillator with eigenvalues $e^{-t(n+1/2)}$, $n\in\mathbb{N}$, and a transverse part involving the shifted wave group $e^{\pm it\sqrt{\Delta-1/4}}$ on $\mathcal{N}$, together with the holomorphic and anti-holomorphic discrete series. The model clarifies two classical links between geodesic dynamics and the Laplace spectrum. Comparing the spectral trace of the propagator in the $X$-adapted Hilbert model with the Atiyah--Bott--Guillemin flat trace gives a dynamical form of the Selberg trace formula: closed geodesics arise from the flat trace, while the spectral side comes from the explicit $SL_{2}(\mathbb{R})$-decomposition. The same factorization also explains the large-time structure of spherical mean operators on $\mathcal{N}$: after the natural $e^{t/2}$-renormalization and the removal of a finite-rank low-energy part, the shifted wave equation on $\mathcal{N}$ emerges as the leading effective dynamics. Thus the construction provides an explicit Hilbert-space structure relating classical geodesic dynamics, Ruelle resonances, and the spectral theory of the surface.