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

Let $$\theta $$ 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 We determine all pairs $$(A_{f,g},A_{g,h})$$ ( A f , g , A g , h ) of generalized weighted quasi-arithmetic means being square iterative roots of $$(A_{F,G},A_{G,H})$$ ( A F , G , A G , H ) , that is, the equation $$( A_{f,g},A_{g,h}) \circ ( A_{f,g},A_{g,h}) =(A_{F,G},A_{G,H}),$$ ( A f , g , A g , h ) ∘ ( A f , g , A g , h ) = ( A F , G , A G , H ) , is solved under three times differentiability of the functions f, g, h, F, G, H. As an application, some special cases are presented. are indicated.

Quantum Variance for Eisenstein Series

In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on $\operatorname{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions.