Quantum ergodicity is a generic property of the eigenstates of quantum systems whose classical counterparts are ergodic. It has been shown for many different systems that, after excluding an insignificant proportion of the exceptional eigenstates, the remaining eigenstates become equidistributed in the high energy limit.
Despite being one of the few questions in quantum chaos that have been mathematically proved in very general settings, the answer to the question of quantum ergodicity remains elusive for quantum graphs. In fact, the question itself is not entirely clear.
In this talk we will attempt to review what is known about the question. The first fundamental fact is that the eignestates of any fixed quantum graph are not quantum ergodic: one needs to take sequences of graphs to see equidistribution. Then we discuss a negative result for a sequence of graphs that is well understood: the star graphs. We show that the eigenstates of the star graphs do not equidistribute.
Finally, we discuss a positive result for a family of quantum graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al. As observables we take the L_2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance.
More precisely, given a one-dimensional, Lebesgue measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs whose vertices correspond to elements of the partitions and whose classical analogues (in the sense of Kottos and Smilansky) are approximating the Perron-Frobenius operator corresponding to the above map. We show that, except possibly for a subsequence of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs.
This talk is based on the joint works with Jon Keating, Uzy Smilansky, Brian Winn.