Semiclassical Structure of Chaotic Resonance Eigenfunctions

We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the nonunitary quantum propagator and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as $\hslash \to 0$ on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates $\{\psi (\hslash ){\}}_{\hslash \to 0}$ is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker’s map, for which the probability density in position space is observed to have self-similarity properties.

Figure 1

Left panel: The average of the Husimi functions of the 100 longest-lived right eigenstates of the baker’s map, for $N=3^7$ (intensity increases from blue to red). Right panel: The corresponding Wigner function average (in the white regions, the function is nonpositive). Note that the $\hslash$ scale is $1/N\approx 0.005$.

Figure 2

The weight $⟨{\Psi }_n^R|{\pi }_m|{\Psi }_n^R⟩$ of right eigenstates of the baker’s map on the regions ${\mathcal{R}}_{+}^m$, for $N=3^6$. The solid line is the semiclassical approximation (9).

Figure 3

Probability density in momentum space for right eigenstates $|⟨p|{\Psi }^R⟩{|}^2$, averaged over the 20 longest-lived states (eigenvalue modulus ranging from 0.90 to 0.83). This function is approximately supported on the Cantor set, as can be seen from a magnification by a factor of 3 (right panel). $N=3^7$ and so the $\hslash$ scale is $1/N\approx 0.005$.

Figure 4

Probability density in position space for right eigenstates $|⟨q|{\Psi }^R⟩{|}^2$, averaged over states with similar decay rates. (b) and (d) are magnifications of (a) and (c). The average eigenvalue modulus is approximately 0.4 for (a),(b) and 0.7 for (c),(d). In both cases, the self-similarity is striking. $N=3^7$.