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 $\varepsilon \frac{ \hbar}{ |\log\hbar|}$
Categories: math.AP math-ph math.DS math.MP