Restricted QUE conjecture for Hecke–Maass cusp forms

Let $u_j$ be a Hecke–Maass cusp form with Laplace eigenvalue $1/4+T_j^2$, Hecke eigenvalues $\lambda_{u_j}(m)$, parity $u_j(-x+iy)=\delta_j u_j(x+iy)$ with $\delta_j\in\{-1,1\}$, and normalization

$$\int_{\Gamma\backslash\mathbb H}|u_j(z)|^2\frac{dx\,dy}{y^2}=\operatorname{vol}(\Gamma\backslash\mathbb H)=\frac{\pi}{3}.$$

Let $\psi$ and the remaining parameters satisfy the conditions of the theorem being analogized, and let $B_q(s,u_j)$ be the finite Euler product specified in the source. Restricted QUE conjecture for Hecke–Maass cusp forms. As $T_j\to\infty$,

$$\int_0^\infty \psi^2(y)|u_j(x+iy)|^2\frac{dy}{y}\sim\int_0^\infty \psi^2(y)\left[1+\delta_j B_q(1,u_j)J_0\left(\frac{\theta T_j}{y}\right)\right]\frac{dy}{y}.$$

This is proposed as an analogue for Maass cusp forms of the paper’s restricted quantum unique ergodicity result for Eisenstein series. The source notes that restricted QUE is more difficult than standard QUE; the status of this precise asymptotic is not resolved in the supplied text.