Power-law diagonal variance conjecture

Let $\Omega$ be the billiard domain, let $\hat{A}$ be multiplication by a test function $A$, and define its spatial average by

$$\overline{A}:=\frac{1}{\operatorname{vol}(\Omega)}\int_\Omega A({\mathbf r})\,{d{\mathbf r}}.$$

For eigenfunctions $\phi_n$ of the Laplacian with eigenvalues $E_n$, define

$$V_A(E):=\frac{1}{N_L(E)}\sum_{n:E_n\in[E,E+L(E)]}\left|\langle\phi_n,\hat{A}\phi_n\rangle-\overline{A}\right|^2,$$

where $N_L(E):=N(E+L(E))-N(E)$ and $N(E):=\#\{n:E_n\leq E\}$. Power-law diagonal variance conjecture. For ergodic flow, as $E\to\infty$, there are constants $a$ and $\gamma$ such that

$$V_A(E)\sim aE^{-\gamma}.$$

This conjecture quantifies the rate at which diagonal matrix elements approach their classical average. The paper tests it numerically; random-wave and Feingold–Peres heuristics predict $\gamma=1/2$, but the general asymptotic law is not established.