On the scarring of eigenstates in some arithmetic hyperbolic manifolds

TL;DR

This paper extends the understanding of quantum ergodicity by showing that in certain three-dimensional hyperbolic manifolds derived from quaternion algebras, eigenstates do not exhibit strong localization on geodesics or totally geodesic surfaces.

Contribution

It generalizes previous results from two-dimensional surfaces to three-dimensional manifolds, demonstrating the absence of strong scarring in this broader class.

Findings

No strong scarring on closed geodesics in the studied manifolds

No strong scarring on Gamma-closed totally geodesic surfaces

Extends quantum ergodicity results to 3D hyperbolic manifolds

Abstract

In this paper, we deal with the conjecture of 'Quantum Unique Ergodicity'. Z. Rudnick and P. Sarnak showed that there is no 'strong scarring' on closed geodesics for arithmetic congruence surfaces derived from a quaternion division algebra. We extend this result to a class of three-dimensional Riemannian manifolds X=Gamma\H^3 that are again derived from quaternion division algebras. We show that there is no 'strong scarring' on closed geodesics or on Gamma-closed imbedded totally geodesics surfaces of X.