Quantum Variance for Hecke–Maass Cusp Forms on the Vertical Geodesic

Let $X=\mathrm{PSL}_2(\mathbb Z)\backslash\mathbb H$, 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 $ψ\in C_c^\infty(0,\infty)$, define $μ_j(ψ)=\int_0^\inftyψ(y)|u_j(iy)|^2\,\frac{dy}{y}, \qquad M(ψ)=\frac6π\int_0^\inftyψ(y)\,\frac{dy}{y}.$ For $T\ge2$, $1\le U\le T$, and every $\varepsilon>0$, we prove $\sum_{\substack{u_j\ {\rm even}\\T<t_j\le T+U}} L(1,\mathrm{sym}^2u_j)|μ_j(ψ)-M(ψ)|^2 \ll_{ψ,\varepsilon}T^{1+\varepsilon}.$ As a consequence, for fixed $0<θ\le1$, $0<η<θ/2$, and every $\varepsilon>0$, if $T^θ\le U\le T$, then all but $O_{ψ,θ,η,\varepsilon} \!\left(T^{1+2η+\varepsilon}\right)$ of the even forms with $T<t_j\le T+U$ satisfy $μ_j(ψ)=M(ψ)+O(T^{-η})$. In particular, the exceptional proportion tends to zero in this short spectral interval.