Quantum Localization in Rough Billiards

We study the level spacing statistics $p\left(s\right)$ and eigenfunction properties in a billiard with a rough boundary. Quantum effects lead to localization of classical diffusion in the angular momentum space and the Shnirelman peak in $p\left(s\right)$ at small $s$. The ergodic regime with Wigner-Dyson statistics is identified as a function of roughness. Applications to $Q$ spoiling in optical resonators are also discussed.