Description
Quantum ergodicity is a deep problem arising in the theory of quantum chaos and number theory. This problem asks the behaviour of Laplace eigenfunctions for chaotic systems in the spectral limit.
For negatively curved compact manifolds, the Quantum Unique Ergodicity (QUE) conjecture of Rudnick and Sarnak pedicts that the mass of Laplace eigenfunctions equidistribute. For arithmetic surfaces (compact or not), this is now a theorem by Lindenstrauss and Soundararajan.
There exist various modifications of quantum ergodicity problems, such as restriction theorems on shrinking balls or submanifolds. For the modular surface, these problems were first studied by Young and Humphries.
In this talk, I will discuss some of our recent results for QUE on shrinking balls on arithmetic hyperbolic manifolds, with an emphasis on 3-surfaces. This talk is based on a work in progress with Robin Frot and Nicole Raulf.