Quantum ergodicity conjecture for resonant states on cusp surfaces

Let $M$ be a surface with cusp ends, and let $\mu_0$ be the Liouville measure. Assume that the geodesic flow on $M$ is ergodic with respect to $\mu_0$. A density-one subsequence of resonant states in any strip

$$\{\operatorname{Im}\lambda>-C\}$$

should converge microlocally to $\mu_0$.

Quantum ergodicity conjecture for resonant states. Under these assumptions, there exists a density-one subsequence of resonant states in every strip $\{\operatorname{Im}\lambda>-C\}$ converging microlocally to the Liouville measure $\mu_0$.

This is the analogue of quantum ergodicity for resonant states when most resonances approach the real axis. The preceding discussion establishes that a density-one subsequence of resonances converges to the real line, but does not establish the asserted microlocal convergence to $\mu_0$; the conjecture remains open in the source.