Abstract
Given any compact hyperbolic surface M, and a closed geodesic on M, we construct of a sequence of quasimodes on M whose microlocal lifts concentrate positive mass on the geodesic. Thus, the quantum unique ergodicity (QUE) property does not hold for these quasimodes. This is analogous to a construction of Faure-Nonnenmacher-De Bievre in the context of quantized cat maps and lends credence to the suggestion that large multiplicities play a role in the known failure of QUE for certain "toy models" of quantum chaos. We moreover conjecture a precise threshold for the order of quasimodes needed for QUE to hold-the result of the present paper shows that this conjecture, if true, is sharp.
Original language | English |
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Pages (from-to) | 11934-11960 |
Number of pages | 27 |
Journal | International Mathematics Research Notices |
Volume | 2015 |
Issue number | 22 |
DOIs | |
State | Published - 2015 |
Bibliographical note
Publisher Copyright:© 2015 The Author(s).
Funding
Funders | Funder number |
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National Science Foundation | |
Directorate for Mathematical and Physical Sciences | 1101596 |