Notes on the Self-Reducibility of the Weil Representation and Higher-Dimensional Quantum Chaos

TL;DR

This paper explores the self-reducibility of the Weil representation to derive sharp estimates for exponential sums, leading to a proof of the Hecke quantum unique ergodicity theorem in higher dimensions.

Contribution

It introduces a novel application of the Weil representation's self-reducibility to quantum chaos, providing new bounds and a proof of quantum ergodicity in higher-dimensional tori.

Findings

Established sharp bounds for exponential sums in higher dimensions.

Proved the Hecke quantum unique ergodicity theorem for generic symplectomorphisms.

Enhanced understanding of quantum chaos in multi-dimensional systems.

Abstract

In these notes we discuss the "self-reducibility property" of the Weil representation. We explain how to use this property to obtain sharp estimates of certain higher-dimensional exponential sums which originate from the theory of quantum chaos. As a result, we obtain the Hecke quantum unique ergodicity theorem for generic linear symplectomorphism $A$ of the torus $T^{2N}=R^{2N}/Z^{2N}.