Feingold–Peres off-diagonal variance conjecture

Let $\phi_n$ be Laplacian eigenfunctions with wavenumbers $k_n$, let $\hat{A}$ be multiplication by a test function, and define the mean wavenumber spacing by

$$\Delta_k(E):=\frac{2\pi}{E^{1/2}\operatorname{vol}(\Omega)}.$$

For a fixed $\omega\in\mathbb{R}$, define

$$V_A(E;\omega):=\frac{\Delta_k(E)}{2\epsilon(E)N_L(E)}\sum_{\substack{m,n:E_n\in[E,E+L(E)]\\|k_m-k_n-\omega|\leq\epsilon(E)}}|\langle\phi_n,\hat{A}\phi_m\rangle|^2.$$

Feingold–Peres off-diagonal variance conjecture. For ergodic flow, as $E\to\infty$,

$$V_A(E;\omega)\sim\frac{\widetilde{C}_A(\omega)}{\operatorname{vol}(\Omega)}E^{-1/2},$$

where $0<\epsilon(E)=O(E^{-1/4})$. This predicts the high-energy variance of off-diagonal matrix elements in a frequency window. It is supported by heuristic arguments and numerical tests, but is not established in the generality stated.