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

Let $X={\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z})\mathrm{\backslash }\mathbb{H}$, and let $\{u_j\}$ be an orthonormal basis of even Hecke–Maass cusp forms on $X$ with Laplace eigenvalues $1\mathrm{/}4+t_j^2$. For a real-valued $\psi \in C_c^{\mathrm{\infty }}(0,\mathrm{\infty })$, define $${\mu }_j(\psi )={\int }_0^{\mathrm{\infty }}\psi (y)\mathrm{\mid }{u}_j(iy){\mathrm{\mid }}^2\frac{dy}{y},M(\psi )=\frac{6}{\pi }{\int }_0^{\mathrm{\infty }}\psi (y)\frac{dy}{y}\mathrm{.}$$ For $T\ge 2$, $1\le U\le T$, and every $\epsilon >0$, we prove $$\sum _{\begin{array}{c}{u}_j\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\\ T<t_j\le T+U\end{array}}L(1,{\mathrm{s}\mathrm{y}\mathrm{m}}^2{u}_j)\mathrm{\mid }{\mu }_j(\psi )-M(\psi ){\mathrm{\mid }}^2{\ll }_{\psi ,\epsilon }{T}^{1+\epsilon }\mathrm{.}$$ As a consequence, for fixed $0<\theta \le 1$, $0<\eta <\theta \mathrm{/}2$, and every $\epsilon >0$, if $T^{\theta }\le U\le T$, then all but $$O_{\psi ,\theta ,\eta ,\epsilon }\text{\hspace{0.17em}⁣}\left(T^{1+2\eta +\epsilon }\right)$$ of the even forms with $T<t_j\le T+U$ satisfy ${\mu }_j(\psi )=M(\psi )+O(T^{-\eta })$. In particular, the exceptional proportion tends to zero in this short spectral interval.