Lior Rosenzweig: Scars for point scatterers on arithmetic tori

Abstract: 

The Seba billiard was introduced to study the transition between integrability and chaos in quantum systems. We investigate the very closely related "toral point scatterer"-model, namely eigenfunctions of the Laplacian perturbed by a delta-potential, on arithmetic 2D and 3D-tori. Recently, Kurlberg and Ueberschaer (2D), following a work of Rudnick and Ueberschaer, and Yesha (3D) proved Quantum Ergodicity results on the set of new eigenvalues, that is that almost all matrix elements corresponding to new eigenvalues converge to the uniform measure on the phase space. This raises the question of Quantum Unique Ergodicity in this context. We will discuss the existence of scars, that is sequence of matrix elements that do not converge to the uniform measure. This is a joint work with Pär Kurlberg