Abstract
AbstractLet $$\theta $$
θ
be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $$\langle |\theta |^2, \varphi \rangle $$
⟨
|
θ
|
2
,
φ
⟩
as $$\varphi $$
φ
traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $$|\theta |^2 \notin L^2$$
|
θ
|
2
∉
L
2
. Applications to the subconvexity, quantum variance and 4-norm problems are indicated.
Funder
Swiss Federal Institute of Technology Zurich
Publisher
Springer Science and Business Media LLC
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