Scarring of quasimodes on hyperbolic manifolds

Abstract

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

arXiv comment28 pages; notations made more uniform and other small fixes