Eisenstein Quasimodes and QUE

TL;DR

This paper investigates Quantum Unique Ergodicity (QUE) for quasimodes on surfaces of constant negative curvature, proving new results for Eisenstein series and extending equidistribution to weaker quasimodes, with implications for arithmetic QUE.

Contribution

The paper proves QUE for Eisenstein series on the modular surface and extends equidistribution results to weaker quasimodes, broadening understanding of quantum ergodicity.

Findings

QUE holds for Eisenstein series on the modular surface.

Equidistribution extends to weaker quasimodes where QUE fails on compact surfaces.

Total mass of limit measures decreases for weaker quasimodes.

Abstract

We consider the question of Quantum Unique Ergodicity for quasimodes on surfaces of constant negative curvature, and conjecture the order of quasimodes that should satisfy QUE. We then show that this conjecture holds for Eisenstein series on the modular surface, extending results of Luo-Sarnak and Jakobson. Moreover, we observe that the equidistribution results of Luo-Sarnak and Jakobson extend to quasimodes of much weaker order--- for which QUE is known to fail on compact surfaces--- though in this scenario the total mass of the limit measures will decrease. We interpret this stronger equidistribution property in the context of arithmetic QUE, in light of recent joint work with E. Lindenstrauss on joint quasimodes.