From geodesic flow to wave dynamics on hyperbolic surfaces
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913167115812864 |
|---|---|
| author | Faure, Frédéric |
| author_facet | Faure, Frédéric |
| contents | 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 $μ=1/4$, 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 $e^{\pm it\sqrt{Δ-1/4}}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28243 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | From geodesic flow to wave dynamics on hyperbolic surfaces Faure, Frédéric Spectral Theory Mathematical Physics Dynamical Systems Representation Theory [2020] Primary 37D20, 58J50, Secondary 22E45, 43A85, 35P25, 11F72 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 $μ=1/4$, 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 $e^{\pm it\sqrt{Δ-1/4}}$ 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. |
| title | From geodesic flow to wave dynamics on hyperbolic surfaces |
| topic | Spectral Theory Mathematical Physics Dynamical Systems Representation Theory [2020] Primary 37D20, 58J50, Secondary 22E45, 43A85, 35P25, 11F72 |
| url | https://arxiv.org/abs/2605.28243 |