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

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 $⟨|\theta {|}^2,\phi ⟩$ as $\phi$ 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.

arXiv comment22 pages; minor changes; to appear in Math. Z