Scarring of quasimodes on hyperbolic manifolds

TL;DR

This paper constructs specific quasimodes on hyperbolic manifolds that strongly concentrate on a submanifold, revealing new insights into quantum ergodicity and invariant measures in semiclassical limits.

Contribution

It explicitly constructs quasimodes with small spectral width that exhibit strong scarring on submanifolds, advancing understanding of quantum limits on hyperbolic manifolds.

Findings

Existence of quasimodes with spectral width at most εħ/|log ħ|

Quasimodes' microlocal lifts converge to measures supported on submanifolds

Any invariant measure on the unit cotangent bundle appears in the ergodic decomposition

Abstract

Let $N$ be a compact hyperbolic manifold, $M\subset N$ an embedded totally geodesic submanifold, and let $-\hbar^2\Delta_{N}$ be the semiclassical Laplace--Beltrami operator. For any $\varepsilon>0$, we explicitly construct families of \emph{quasimodes} of spectral width at most $\varepsilon\frac{\hbar}{|\log\hbar|}$ which exhibit a "strong scar" on $M$ in that their microlocal lifts converge weakly to a probability measure which places positive weight on $S^*M$ ($\hookrightarrow S^*N$). An immediate corollary is that \emph{any} invariant measure on $S^*N$ occurs in the ergodic decomposition of the semiclassical limit of certain quasimodes of width $\varepsilon \frac{\hbar}{|\log\hbar|}$