On the resonance eigenstates of an open quantum baker map

Details On the resonance eigenstates of an open quantum baker map On the resonance eigenstates of an open quantum baker map

2008-06-10

arxiv.org/abs/0806.1678v2

Nonlinearity 21, 2591 (2008)

Nonlinear Sciences

Chaotic Dynamics

32 pages, 2 figures

Scientific paper

10.1088/0951-7715/21/11/007

We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, $|z_{min}|\leq |z|\leq |z_{max}|$. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius $r$. We prove that, if the moduli converge to $r=|z_{max}|$, then the sequence of eigenstates converges to a fixed phase space measure $\rho_{max}$. The same holds for sequences with eigenvalue moduli converging to $|z_{min}|$, with a different limit measure $\rho_{min}$. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius $|z_{min}|< r < |z_{max}|$, we identify families of eigenstates with precise self-similar properties.

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