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Quantum Variance for Hecke--Maass Cusp Forms on the Vertical Geodesic
Abstract
Let X=PSL2(ℤ)backslashℍ, and let {uj} be an orthonormal basis of even Hecke--Maass cusp forms on X with Laplace eigenvalues 1/4+tj2. For a real-valued ψ∈ Cc∞(0,∞), define μj(ψ)=∫0∞ψ(y)|uj(iy)|2 dy/y, M(ψ)=6/π∫0∞ψ(y) dy/y. For T≥2, 1≤ U≤ T, and every ε>0, we prove ∑substackuj even T<tj≤ T+U L(1,sym2uj)|μj(ψ)−M(ψ)|2 ≪ψ,εT1+ε. As a consequence, for fixed 0<θ≤1, 0<η<θ/2, and every ε>0, if Tθ≤ U≤ T, then all but Oψ,θ,η,ε (T1+2η+ε) of the even forms with T<tj≤ T+U satisfy μj(ψ)=M(ψ)+O(T−η). In particular, the exceptional proportion tends to zero in this short spectral interval.
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BibTeX
@misc{sun2026quantum,
title = {{Quantum Variance for Hecke--Maass Cusp Forms on the Vertical Geodesic}},
author = {Qingfeng Sun and Qizhi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27454},
eprint = {2610.27454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27454},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Variance for Hecke--Maass Cusp Forms on the Vertical Geodesic
AU - Qingfeng Sun
AU - Qizhi Zhang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27454
UR - https://arxiv.org/abs/2610.27454
AB - Let X=PSL_2(ℤ)backslashℍ, and let {u_j} be an orthonormal basis of even Hecke--Maass cusp forms on X with Laplace eigenvalues 1/4+t_j^2. For a real-valued ψ∈ C_c^∞(0,∞), define μ_j(ψ)=∫_0^∞ψ(y)|u_j(iy)|^2 dy/y, M(ψ)=6/π∫_0^∞ψ(y) dy/y. For T≥2, 1≤ U≤ T, and every ε>0, we prove ∑_substacku_j even T<t_j≤ T+U L(1,sym^2u_j)|μ_j(ψ)−M(ψ)|^2 ≪_ψ,εT^1+ε. As a consequence, for fixed 0<θ≤1, 0<η<θ/2, and every ε>0, if T^θ≤ U≤ T, then all but O_ψ,θ,η,ε (T^1+2η+ε) of the even forms with T<t_j≤ T+U satisfy μ_j(ψ)=M(ψ)+O(T^−η). In particular, the exceptional proportion tends to zero in this short spectral interval.
ER -