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Quantum ergodicity for graphs related to interval maps. (English) Zbl 1122.81037

Summary: 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 P. Pakónski, K. Zyczkowski and M. Kus [J. Phys. A, Math. Gen. 34, No. 43, 9303–9317 (2001; Zbl 0995.82040)]. 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.

MSC:

81Q50 Quantum chaos
37A25 Ergodicity, mixing, rates of mixing
37E05 Dynamical systems involving maps of the interval
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
37E25 Dynamical systems involving maps of trees and graphs

Citations:

Zbl 0995.82040
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References:

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