Patterson-Sullivan and Wigner distributions of convex-cocompact hyperbolic surfaces

TL;DR

The paper proves that on convex-cocompact hyperbolic surfaces, the Wigner distributions of unbounded quantum resonances and the Patterson–Sullivan distributions are asymptotically identical up to a universal complex constant, extending known results from compact to non-compact settings. The authors leverage the quantum–classical correspondence, Helgason boundary values, and the (weighted) Radon transform, together with stationary-phase analysis of intertwining operators, to relate quantum microlocal lifts to classical boundary data. A precise asymptotic relation of the form $W_{\phi,\phi'}(u) = c\, r^{-1/2}( \mathrm{PS}_{\phi,\phi'}(u) + O(r^{-1}) )$ with $c = (\sqrt{\pi})^{-1} e^{-i\pi/4}$ is established for resonances with large imaginary part and bounded real parts. This generalizes AZ07 and HHS12 to Schottky and other convex-cocompact surfaces, reinforcing the quantum–classical correspondence in open hyperbolic manifolds and informing quantum ergodicity and scattering theory on non-compact spaces.

Abstract

We prove that the Patterson-Sullivan and Wigner distributions on the unit sphere bundle of a convex-cocompact hyperbolic surface are asymptotically identical. This generalizes results in the compact case by Anantharaman-Zelditch and Hansen-Hilgert-Schröder.

Figures (3)

• Figure 1: Left: Fictional plot of quantum resonances of a non-elementary convex-cocompact hyperbolic surface that might look like the one illustrated in the picture, for example. If the Jakobson-Naud conjecture holds for this surface, then any strip of positive width (indicated in blue) at the left of the line $\mathop{\mathrm{Re}}\nolimits s=\frac{\delta}{2}$ contains an unbounded sequence of resonances. Independently of the Jakobson-Naud conjecture, the large gray strip of width $>\frac{3}{2}$ at the left of the critical line $\mathop{\mathrm{Re}}\nolimits s=\frac{1}{2}$ always contains an unbounded sequence of resonances. Right: The elementary case of a hyperbolic cylinder. Here the quantum resonances lie on a lattice.
• Figure 2: Plotted are numerically calculated quantum resonances $s_0\in \mathbb{C}$ for a $3$-funneled Schottky surface, following the approach of Weich2015_ResonanceChainsBorthwickbook.
• Figure 3: The projection ${\mathscr C}K\in \mathbb H^2=G/K$ of a compact set ${\mathscr C}\subset G$ and a geodesic in $\mathbb H^2$ passing through ${\mathscr C}K$ with its endpoints $b,b'\in \partial_\infty\mathbb H^2=\mathbb{S}^1$. This particular geodesic realizes the minimal distance in $\mathbb{S}^1$ of endpoints of geodesics passing through ${\mathscr C}K$ -- for any other such geodesic, its endpoints will be at least as far apart as $b$ and $b'$. In particular, the pairs of such endpoints cannot be arbitrarily close to the diagonal in $\partial_\infty\mathbb H^2\times \partial_\infty\mathbb H^2$.

Theorems & Definitions (26)

• Theorem 1
• Lemma 2.1
• proof
• Definition 2.2
• Remark 2.1: Conventions
• Proposition 2.3
• Example 2.4: Schottky surfaces
• Definition 2.5
• Lemma 2.6
• proof