Strong quantitative quantum ergodicity conjecture for Maass-Hecke cusp forms

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Let ϕ\phi be a Maass-Hecke cusp form with spectral parameter tϕt_\phi, Fourier coefficients ρϕ(n)\rho_\phi(n), and let δ0,m\delta_{0,m} denote the Kronecker delta. For ψC0(0,)\psi\in C_0^\infty(0,\infty), write ψWk,(0,)||\psi||_{W^{k,\infty}(0,\infty)} for its Sobolev norm. Strong QQUE conjecture. There exist ν>0\nu>0 and k<k<\infty such that, for any ψC0(0,)\psi\in C_0^\infty(0,\infty),

1tϕnρϕ(n+m)ρϕ(n)ψ(πntϕ)8πδ0,m0ψ(y)dymtϕνψWk,(0,).\left|\frac{1}{t_\phi}\sum_n \rho_\phi(n+m)\rho_\phi(n) \psi\left(\frac{\pi|n|}{t_\phi}\right) - \frac{8}{\pi} \delta_{0,m}\int_0^\infty \psi(y)dy\right| \ll_m t_\phi^{-\nu}||\psi||_{W^{k,\infty}(0,\infty)}.

This is the strong rate-of-convergence form of arithmetic quantum unique ergodicity for the Fourier coefficients of Maass-Hecke cusp forms. The source presents it as the strong form of quantitative QUE, while the supplied text gives no resolution evidence, so its status remains open.

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Primary source

Junehyuk Jung, “Quantitative quantum ergodicity and the nodal domains of Maass-Hecke cusp forms”, arXiv:1301.6211 (2016).

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