## Abstract

We consider spaces of high-energy quasimodes for the Laplacian on a compact hyperbolic surface, and show that when the spaces are large enough, one can find quasimodes that exhibit strong localization phenomena. Namely, take any constant c, and a sequence of cr_{j}-dimensional spaces S_{j} of quasimodes, where 1/4 + r_{j}^{2} → ∞ is an approximate eigenvalue for S_{j}. Then we can find a sequence of vectors ψ_{j} ∈ S_{j}, such that any weak-* limit point of the microlocal lifts of {pipe}ψ_{j}{pipe}^{2} localizes a positive proportion of its mass on a singular set of codimension 1. This result is sharp, in light of the QUE result of [BL12] for certain joint quasimodes that include spaces of size o(r_{j}), with arbitrarily slow decay.

Original language | English |
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Pages (from-to) | 393-417 |

Number of pages | 25 |

Journal | Israel Journal of Mathematics |

Volume | 198 |

Issue number | 1 |

DOIs | |

State | Published - Nov 2013 |

Externally published | Yes |