Long Time Quantum Evolution of Observables

TL;DR

This paper develops a semi-classical quantization method for hyperbolic cusp manifolds, proving Egorov's lemma up to Ehrenfest times and establishing a version of Quantum Unique Ergodicity for Eisenstein series.

Contribution

It introduces a semi-classical quantization adapted to hyperbolic cusp manifolds and extends Quantum Unique Ergodicity results to Eisenstein series near the unitary axis.

Findings

Proved Egorov lemma up to Ehrenfest times on hyperbolic cusp manifolds

Established Quantum Unique Ergodicity for Eisenstein series near the unitary axis

Developed a geometrically adapted semi-classical quantization procedure

Abstract

We build a semi-classical quantization procedure for finite volume man- ifolds with hyperbolic cusps, adapted to a geometrical class of symbols. We prove an Egorov Lemma until Ehrenfest times on such manifolds. Then we give a version of Quantum Unique Ergodicity for the Eisenstein series for values of s converging slowly to the unitary axis.