Scarring of quasimodes on hyperbolic manifolds

作者：Eswarathasan Suresh; Silberman Lior

来源：Nonlinearity, 2018, 31(1): 1-29.

DOI：10.1088/1361-6544/aa92e3

• Laplace-Beltrami operator
• eigenfunction asymptotics
• invariant measures
• hyperbolic geometry
• quantum unique ergodicity

Let N be a compact hyperbolic manifold, M subset of N an embedded totally geodesic submanifold, and let -h(2)Delta(N) be the semiclassical Laplace-Beltrami operator.
For any epsilon > 0 we explicitly construct families of quasimodes of energy width at most epsilon h/vertical bar log h vertical bar 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 (hooked right arrow S*N). An immediate corollary is that any invariant measure on S*N occurs in the ergodic decomposition of the semiclassical limit of certain quasimodes of width epsilon h/vertical bar log h vertical bar.

• 出版日期2018-1