Physical-Space Scarring in Generic Bunimovich Stadia

Abstract

For the family of Dirichlet stadia $S_t$ whose rectangular part has height $\pi$ and half-length $\pi t/2$, $t \in [1,2]$, we show that for Lebesgue almost every $t$ there exist real eigenfunctions $u_j$ and a smooth mean-zero physical observable $Q$ for which $\langle Q u_j,u_j\rangle$ has a non-zero subsequential limit. Consequently, along the same subsequence, the eigenfunction mass fails to equidistribute on a fixed region whose relative boundary in the interior of the stadium is smooth. This proves a physical-space strengthening of Hassell's non-QUE theorem for generic stadia, and thus gives an affirmative answer to Tao's question in Hassell's generic setting. The proof uses the classification of generic stadia in Hassell's argument. In each of the resulting cases, we construct an appropriate physical observable $Q$ that converts Hassell's phase-space obstruction to QUE into physical-space non-equidistribution.