On the resonance eigenstates of an open quantum baker map

Abstract

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, $|z_{min}|\le |z|\le |z_{max}|$. 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.

32 pages, 2 figures

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• Statistics of eigenfunctions in open chaotic systems: a perturbative approach
• Formation and interaction of resonance chains in the open 3-disk system
• Structure of resonance eigenfunctions for chaotic systems with partial escape
• Origin of the enhancement of tunneling probability in the nearly integrable system
• Universal intensity statistics of multifractal resonance states
• Chaotic resonance modes in dielectric cavities: Product of conditionally invariant measure and universal fluctuations
• Transient features of quantum open maps
• Asymptotic mean density of sub-unitary ensemble
• Local random vector model for semiclassical fractal structure of chaotic resonance states
• Resonance spectra for quantum maps of kicked scattering systems by complex scaling
• Left-Right Husimi Representation of Chaotic Resonance States: Invariance and Factorization