The spectral decomposition of $|\theta|^2$

TL;DR

This paper develops a spectral decomposition for the squared modulus of elementary theta functions, providing new asymptotic formulas with applications to subconvexity, quantum variance, and norm problems.

Contribution

It introduces a novel spectral decomposition for $| heta|^2$, even though it generally does not belong to $L^2$, and derives strong asymptotic formulas for its inner products.

Findings

Established spectral decomposition for $| heta|^2$

Derived strong asymptotic formulas for inner products

Indicated applications to subconvexity and quantum variance

Abstract

Let $\theta$ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $\langle |\theta|^2, \varphi \rangle$ as $\varphi$ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $|\theta|^2 \notin L^2$. Applications to the subconvexity, quantum variance and $4$-norm problems are indicated.