Fractal Weyl law behavior in an open Hamiltonian system

We numerically show fractal Weyl law behavior in an open Hamiltonian system that is described by a smooth potential and which supports numerous above-barrier resonances. This behavior holds even relatively far away from the classical limit. The complex resonance wave functions are found to be localized on the fractal classical repeller.

Figure 1

(Color online) Superposition of the two branches of the repeller computed as described in the text and projected onto the SOS defined by $x=0,\dot{x}<0$ and $E=1.8E_s$. The bounding curve, solid black line (red online), of the SOS is also shown. Both axes have been scaled to the interval (0,1) to allow for comparison with the Husimi plot of Fig. 3.

Figure 2

Frame (a) shows resonance positions $\left(E_r\right)$ and widths $\left(\Gamma \right)$ (scaled by the saddle-point energy $E_s$) for $\hslash =1$. In (b) the classical quantity $\text{ln}{C}_2\left(s\right)$ is shown vs $\text{ln}s$ together with a best fit to a line whose slope is the correlation dimension [see Eq. (2)], i.e., $d_2=1.442\pm 0.008$. Frame (c) shows a best fit to box counted quantum resonances with slope $d=1.231\pm 0.028$. The quantum resonances scale as ${\hslash }^{-m/2}$ or ${\hslash }^{-1.22}$ [5], where $m=\left(1+d_2\right)$ is the dimension of the trapped set; the quantum box counting dimension is 1.23. Despite considering a relatively narrow range of $\hslash$, lying far above the classical limit, the classical and quantum fractal dimensions agree well.

Figure 3

(Color online) Contours of the averaged Husimi function calculated as described in the text superimposed on a representation of the classical repeller. Twenty resonance states on each side of $E=1.8E_s$ were included. The points which represent the classical repeller on the SOS have been kernel smoothed which, in essence, assigns a local density of points and then colors that section of the plot accordingly [3]—compare to Fig. 1. The color scale runs from white through light gray (yellow and light blue online) to dark gray (dark blue online) and represents the density from low to high accordingly.