Quantum Ergodicity vs. Superscars in Seba's Billiard

Shnirelman discovered in the 1970s that the eigenfunctions of the Laplacian on a compact Riemannian manifold whose flow is ergodic with respect to Liouville measure exhibit an analogue of classical ergodicity at the quantum level. This phenomenon became known as "Quantum Ergodicity" and Schnirelman's proof was completed by Zelditch and Colin de Verdiere in the 1980s. Following a brief introduction to the subject, I will show that Quantum Ergodicity can also hold in systems which are essentially integrable, provided they have some arithmetic structure. Finally, in the absence of such an arithmetic structure, a very different phenomenon occurs: scarring. This talk is based on joint work with Par Kurlberg, KTH Stockholm.