Weyl's law and quantum ergodicity for maps with divided phase space

Details Weyl's law and quantum ergodicity for maps with divided phase space Weyl's law and quantum ergodicity for maps with divided phase space

2004-04-20

arxiv.org/abs/nlin/0404038v2

Nonlinearity 18 (2005) 277-304

Nonlinear Sciences

Chaotic Dynamics

Colour figures. Black & white figures available at http://www2.maths.bris.ac.uk/~majm. Appendix by Steve Zelditch

Scientific paper

For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one ergodic component, and is periodic on the remaining domains, we prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the equidistribution of eigenfunctions with respect to the ergodic component of the classical map (quantum ergodicity). We apply our main theorems to quantised linked twist maps on the torus. In the Appendix, S. Zelditch connects these studies to some earlier results on `pimpled spheres' in the setting of Riemannian manifolds. The common feature is a divided phase space with a periodic component.

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