Renormalization of Quantum Anosov Maps: Reduction to Fixed Boundary Conditions

Details Renormalization of Quantum Anosov Maps: Reduction to Fixed Boundary Conditions Renormalization of Quantum Anosov Maps: Reduction to Fixed Boundary Conditions

2000-05-18

arxiv.org/abs/nlin/0005038v1

Nonlinear Sciences

Chaotic Dynamics

REVTEX, 12 pages, 1 table. To appear in Physical Review Letters

Scientific paper

10.1103/PhysRevLett.84.5994

A renormalization scheme is introduced to study quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number ($k$) is always finite. It is shown that the quasienergy eigenvalue problem of a QAM for {\em all} $k$ BCs is exactly equivalent to that of the renormalized QAM (with Planck's constant $\hbar ^{\prime}=\hbar /k$) at some {\em fixed} BCs that can be of four types. The quantum cat maps are, up to time reversal, fixed points of the renormalization transformation. Several results at fixed BCs, in particular the existence of a complete basis of ``crystalline'' eigenstates in a classical limit, can then be derived and understood in a simple and transparent way in the general-BCs framework.

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