Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Tuesday, 02.05.2006 16 c.t.

Quantum ergodicity on graphs related to 1d interval maps

by Prof. Dr. Gregory Berkolaiko
from Texas A&M University, College Station, USA

Contact person: Holger Schanz


Seminarraum Haus 2, 4. Stock (Bunsenstr.)


We start by describing the property of the eigenstates of a classicaly chaotic quantum system to equidistribute over the accessible phase-space. This property is known as quantum ergodicity, first formulated in an article by Shnirelman. We prove quantum ergodicity 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.

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