Ergodic properties of quantized toral automorphisms

Journal Article · Wed Jan 01 04:00:00 EST 1997 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.531835· OSTI ID:435141

Klimek, S ^[1]; Lesniewski, A; Maitra, N; Rubin, R ^[2]

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1. Department of Mathematics, IUPUI, Indianapolis, Indiana 46205 (United States)
2. Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138 (United States)

We study the ergodic properties for a class of quantized toral automorphisms, namely the cat and Kronecker maps. The present work uses and extends the results of Klimek and Le{acute s}niewski [Ann. Phys. {bold 244}, 173{endash}198 (1996)]. We show that quantized cat maps are strongly mixing, while Kronecker maps are ergodic and nonmixing. We also study the structure of these quantum maps and show that they are effected by unitary endomorphisms of a suitable vector bundle over a torus. This allows us to exhibit explicit relations between our Toeplitz quantization and the semiclassical quantization of cat maps proposed by Hannay and Berry [Physica D {bold 1}, 267{endash}290 (1980)]. {copyright} {ital 1997 American Institute of Physics.}

• OSTI ID: 435141
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 38; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
DYNAMICS
EIGENSTATES
ERGODIC HYPOTHESIS
QUANTIZATION
QUANTUM MECHANICS
QUANTUM OPERATORS
SEMICLASSICAL APPROXIMATION
VECTORS