The quantum unique ergodicity conjecture for thin sets

TL;DR

This paper investigates analogs of the quantum unique ergodicity conjecture for specific geometric sets and proves the conjecture for Eisenstein series on a particular geodesic in the modular surface.

Contribution

It establishes the QUE conjecture for Eisenstein series restricted to the geodesic connecting 0 and infinity on the modular surface.

Findings

Proved QUE for Eisenstein series on a specific geodesic

Extended QUE analogs to geodesics and horocycles

Analyzed shrinking families of sets in quantum ergodicity context

Abstract

We consider some analogs of the quantum unique ergodicity conjecture for geodesics, horocycles, or ``shrinking'' families of sets. In particular, we prove the analog of the QUE conjecture for Eisenstein series restricted to the infinite geodesic connecting 0 and infinity inside the modular surface.