[Submitted on 27 May 2026]

View PDF

Abstract: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.

Submission history

From: Frederic Faure [view email]
[v1] Wed, 27 May 2026 09:57:12 UTC (735 KB)

Current browse context:

math.SP

Bookmark

Bibliographic and Citation Tools

Code, Data and Media Associated with this Article

Demos

Recommenders and Search Tools

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.