Weyl law for open systems with sharply divided mixed phase space

A generalization of the Weyl law to systems with a sharply divided mixed phase space is proposed. The ansatz is composed of the usual Weyl term which counts the number of states in regular islands and a term associated with sticky regions in phase space. For a piecewise linear map, we numerically check the validity of our hypothesis, and find good agreement not only for the case with a sharply divided phase space but also for the case where tiny island chains surround the main regular island. For the latter case, a nontrivial power law exponent appears in the survival probability of classical escaping orbits, which may provide a clue to develop the Weyl law for more generic mixed systems.

Figure 1

Phase space of the map (1) for (a) $k=2.0$ [corresponding to $n=2$ in Eq. (3)] and (b) $k=3.0$ $(n=3)$. For these values, no hierarchical structures appear in phase space. Empty white regions are composed of periodic orbits. For (c) $k=4.0$, the regular region shrinks to a family of stable periodic orbits. The case (d) $k=2.5$ does not satisfy the condition (3), and tiny hierarchical islands around the regular region emerge.

Figure 2

The set of surviving points $({\theta }_n,p_n)$ for $k=3.0$, after (a) $n=50$ iterations and (b) $n=10$ iterations, where $2\times {10}^5$ initial points are distributed uniformly in the region $(\theta ,p)\in [0,1)\times [-0.5,0.5]$ but outside the regular region.

Figure 3

Survival probability in the chaotic part in phase space, for the case $k=3.0$. (a) Normal-log and (b) log-log plot. $2^{20}$ points are initially distributed uniformly over phase space, and the survival probability is obtained by counting the number of orbits remaining in $R$, up to ${10}^4$ steps. Note that in panel (a), we observe exponential decay up to $\tau \lesssim 20$, while in panel (b) the decay follows a power law for $\tau \gtrsim 100$.

Figure 4

(a) The set of surviving points $({\theta }_n,p_n)$ for $k=4.0$ after $n=50$ iterations, with ${10}^6$ initial points uniformly distributed over $R$. (b) Survival probability for $k=4.0$.

Figure 5

(a) Phase space portrait for $k=3.0$ and smoothed potential (7) with $M=5$. The blue line represents the boundary between regular and chaotic region in case of $M=\infty$. (b) Magnification around the boundary. Small islands appear due to the smoothing of the potential function.

Figure 6

Survival probability of the chaotic orbits for the smoothed potential (7) with $M=100$ and (a) $k=3.0$, (b) $k=4.0$.

Figure 7

(a) Quantum decay rates $\Gamma$ for $k=3.0$ and $N=4096$. The values of $\Gamma$ are arranged in the ascending order. (b) Four typical resonance states in the Husimi representation. The numeric labels indicate the corresponding decay rate in panel (a).

Figure 8

(a) Quantum decay rates $\Gamma$ for $k=4.0$ and $N=4096$, arranged in the ascending order. (b) Three typical resonance states in the Husimi representation. The labels again correspond to the indicated decay rates in panel (a).

Figure 9

$W_{\gamma }\left(\hslash \right)/N$ vs $\gamma$ for (a) $k=3.0$ and (b) $k=4.0$. The black dashed line expresses the formula (9). The curves correspond to $N=512$ (red), 1024 (light green), 2048 (dark blue), 4096 (pink), 8192 (light blue), 16384 (orange).

Figure 10

Plot of $W_{\alpha \hslash }^{\mathrm{sticky}}\left(\hslash \right)\frac{2\pi }{C\alpha }$ as a function of $1/\hslash$ for (a) $k=3.0$ and (c) $k=4.0$. We put $\alpha =100$ for $k=3.0$ and $\alpha =200$ for $k=4.0$. $W_{\alpha \hslash }^{\mathrm{sticky}}\left(\hslash \right)\frac{2\pi }{C\alpha }$ as a function of the parameter $\alpha$ for (b) $k=3.0$ and (d) $k=4.0$, respectively. For each $\alpha$, we have calculated several values of $N$ (red curve) where $N$ is chosen such that $\gamma \lesssim 0.05$ is satisfied. The blue line shows the average over $N$. If the ansatz (12) holds, the function $W_{\alpha \hslash }^{\mathrm{sticky}}\left(\hslash \right)\frac{2\pi }{C\alpha }$ should tend to the green line.

Figure 11

Plot of $W_{\alpha \hslash }^{\mathrm{sticky}}\left(\hslash \right)$ as a function of $1/\hslash$ for $k=3.0$ and $\alpha =100$. The green line represents $W_{\alpha \hslash }^{\mathrm{sticky}}\simeq 0.27{\hslash }^{-0.5}$.

Figure 12

For $k=3.0$, the assumed function $C{(\alpha /6\pi )}^{0.5}{N}^{0.5}$ is fitted to determine (a) $C$ and (b) $\nu$.

Figure 13

Plot of $W_{\alpha \hslash }^{\mathrm{sticky}}\left(\hslash \right)$ as a function of $1/\hslash$ for $k=4.0$ and $\alpha =100$. The green line represents $W^{\mathrm{sticky}}\simeq 0.34{\hslash }^{-0.5}$.

Figure 14

For $k=4.0$, the assumed function $C{(\alpha /6\pi )}^{0.5}{N}^{0.5}$ is fitted to determine (a) $C$ and (b) $\nu$.