Scarring of quasimodes on hyperbolic manifolds

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 $\varepsilon \frac{\hbar}{|\log\hbar|}$