Scarring of quasimodes on hyperbolic manifolds

Lior Silberman, Suresh Eswarathasan

Abstract:

Let N be a compact hyperbolic manifold, M⊂N an embedded totally geodesic submanifold, and let −ℏ2ΔN be the semiclassical Laplace--Beltrami operator.
For any ε>0, we explicitly construct families of \emph{quasimodes} of spectral width at most εℏ|logℏ| 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 (↪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 εℏ|logℏ|