Graph Eigenfunctions and Quantum Unique Ergodicity

TL;DR

This paper investigates the behavior of joint eigenfunctions on certain geometric surfaces, demonstrating that their quantum limit measures exhibit positive entropy, leading to results on quantum unique ergodicity.

Contribution

It extends previous techniques to show quantum unique ergodicity for joint eigenfunctions on compact congruence surfaces and quotients of hyperbolic space.

Findings

Quantum limit measures have positive entropy on almost every ergodic component.

Quantum unique ergodicity is established for these eigenfunctions.

The methods connect eigenfunction behavior with ergodic properties of the underlying surfaces.

Abstract

We apply the techniques of our previous paper to study joint eigenfunctions of the Laplacian and one Hecke operator on compact congruence surfaces, and joint eigenfunctions of the two partial Laplacians on compact quotients of $\mathbb{H}\times\mathbb{H}$. In both cases, we show that quantum limit measures of such sequences of eigenfunctions carry positive entropy on almost every ergodic component. Together with prior work of the second named author, this implies Quantum Unique Ergodicity for such functions.