Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

Abstract

We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and if H ⊂ M is a smooth hypersurface satisfying a generic microlocal asymmetry condition, then restrictions ϕ j | H of an orthonormal basis {ϕ j } of ∆-eigenfunctions of (M, g) to H are quantum ergodic on H. The condition on H is satisfied by geodesic circles, closed horocycles and generic closed geodesics on a hyperbolic surface. A key step in the proof is that matrix elements F ϕ j , ϕ j of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.

Sign Changes of the Eisenstein Series on the Critical Line

“…Assume further that {u j } j=1,2,... is an eigenbasis for the space of even L 2 functions. In this case, the Quantum Ergodic Restriction theorem [CTZ13,DZ13,TZ13] implies that there exists a density one subset B of N such that for any compactly supported smooth function f ∈ C If QUE for the restriction to H is true, then one can take B = N.…”

Local and global analysis of nodal sets

“…We begin with (i). In [TZ13], a geodesic asymmetry condition on a hypersurface was introduced which is sufficient that restrictions of quantum ergodic eigenfunctions on M remain quantum ergodic on the hypersurface. It turns out that the same asymmetry condition plus a flow-out condition implies that a hypersurface is good for a density one subsequence of eigenfunctions and that for any δ > 0, the L 2 norms of the restricted eigenfunctions have a uniform lower bound C δ > 0 for a subsequence of density 1 − δ.…”

Completeness of boundary traces of eigenfunctions

“…For a general hypersurface, there are two components to the Cauchy data, and quantum ergodicity refers to the pair. In [TZ2], the Quantum Ergodic Restriction (QER) is proved for the individual Dirichlet and Neumann if the hypersurface satisfies an asymmetry condition with respect to the geodesic flow. This condition is not needed in Theorem 16 to prove that Dirichlet and Neumann data are individually complete.…”