The logarithmic quasimode threshold conjecture for QUE on hyperbolic surfaces

Let $M$ be a compact hyperbolic surface, and let $\{\psi_j\}$ be a sequence of quasimodes with approximate spectral parameters $r_j\to\infty$. Assume their quasimode widths satisfy $\omega(r_j)=o\!\left(\frac{1}{\log r_j}\right)$.

Logarithmic quasimode threshold conjecture. Then the sequence $\{\psi_j\}$ satisfies Quantum Unique Ergodicity.

The paper constructs quasimodes at logarithmic-scale widths whose microlocal lifts concentrate positive mass on a closed geodesic, so the conjectured threshold would be sharp. It identifies the scale $1/\log r$ as the boundary between possible non-equidistribution and QUE for quasimodes.