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

References in corpus (2)

• Fractal Weyl laws for chaotic open systems
• Quantum-to-classical crossover of quasi-bound states in open quantum systems