On the resonance eigenstates of an open quantum baker map

TL;DR

This paper investigates the resonance eigenstates of an open quantum baker map, showing how their semiclassical limits relate to fractal trapped sets and identifying self-similar properties for eigenstates with eigenvalues of intermediate modulus.

Contribution

It provides a detailed analysis of the semiclassical limits of resonance eigenstates in an open quantum baker map, linking eigenvalue moduli to specific fractal measures and self-similar eigenstate structures.

Findings

Eigenstates with eigenvalue modulus approaching |z_max| converge to a measure supported on a fractal trapped set.

Eigenstates with eigenvalue modulus approaching |z_min| converge to a different fractal-supported measure.

Eigenstates with intermediate eigenvalue moduli exhibit self-similar properties.

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