M . E . & T . W . Arithmetic Quantum Unique Ergodicity for Γ \ H February 4 , 2010

Quantum unique ergodicity – QUE – is concerned with the distributional properties of high-frequency eigenfunctions of the Laplacian on a domain Ω, that is of solutions to the equation ∆φ j + λ j φ j = 0 with the Dirichlet boundary conditions φ j | ∂Ω = 0 and normalization Ω φ 2 j dx dy = 1, where ∆ is the appropriate Laplacian. There is a connection between the high-frequency states and the classical Hamiltonian dynamical system obtained by letting a billiard ball move inside Ω at unit speed, and bouncing off the boundary (if there is a boundary) of Ω with angle of incidence equal to the angle of reflection. If the motion is integrable (for example, if Ω is a circle), then there are invariant sets with measure strictly between 0 and 1 (for example, if Ω is a circle, then the set of orbits tangent to a given concentric circle with radius in a given interval is an invariant set). At first the relationship between high-frequency eigenfunctions and distributional or ergodic properties may be surprising, but in the integrable case we can find eigenfunctions of the Lapla-cian that are highly localized on non-trivial invariant sets (see the survey [1, Figs. 1 & 2]). Even for the simplest of domains Ω a diversity of possibilities occurs. A special case to which ergodic methods may be applied comes from homogeneous dynamics – that is, to actions of subgroups of a Lie group G on quotients Γ \G of finite volume (see [12], [11] for examples and background). In this case quite strong conjectures were made by Rudnick and Sarnak [24].