Quantum ergodicity for graphs related to interval maps

TL;DR

This paper demonstrates quantum ergodicity for a family of graphs derived from ergodic interval maps, showing eigenstate equidistribution and classical-quantum correspondence in the large graph limit.

Contribution

It introduces a method to prove quantum ergodicity for graphs associated with interval maps using periodic orbit expansions and extends the Egorov property to these graphs.

Findings

Eigenstates of the graphs become equidistributed as graphs grow large.

The Egorov property holds for a class of observables in these quantum graphs.

The approach links classical interval dynamics with quantum graph behavior.

Abstract

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). 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. Specifically, 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 directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.