Semiclassical quantization of the diamagnetic hydrogen atom with near–action-degenerate periodic-orbit bunches

The existence of periodic orbit bunches is proven for the diamagnetic Kepler problem. Members of each bunch are reconnected differently at self-encounters in phase space but have nearly equal classical action and stability parameters. Orbits can be grouped already on the level of the symbolic dynamics by application of appropriate reconnection rules to the symbolic code in the ternary alphabet. The periodic orbit bunches can significantly improve the efficiency of semiclassical quantization methods for classically chaotic systems, which suffer from the exponential proliferation of orbits. For the diamagnetic hydrogen atom the use of one or few representatives of a periodic orbit bunch in Gutzwiller’s trace formula allows for the computation of semiclassical spectra with a classical data set reduced by up to a factor of 20.

Figure 1

(Color online) In the multishooting algorithm each orbit is split into short segments starting and ending on one of the axes. The concatenation of various types of segments is guided by the symbolic code of the orbit.

Figure 2

(Color online) Example of a periodic orbit displayed in (a) the full coordinate plane, (b) the first quadrant, and (c) the fundamental domain.

Figure 3

(Color online) Example of a periodic-orbit bunch consisting of 16 near–action-degenerate orbits drawn in the fundamental domain of the $\left(\mu ,\nu \right)$ coordinate space for scaled energy $\tilde{E}=0.5$. The similarities between orbits are obvious in the overview.

Figure 4

(Color online) Example of a near–action-degenerate periodic-orbit bunch at scaled energy $\tilde{E}=0.5$ consisting of 16 orbits with cycle length $L=10$. The trajectories are drawn in semiparabolic $\left(\mu ,\nu \right)$ coordinates and are labeled by the symbolic code. See text for discussion.

Figure 5

(Color online) Sketch of the four reconnection rules. In the $\left(\mu ,\nu \right)$ coordinate space (0) stretches indicated by red solid lines are located in the vicinity of the $\mu =\nu$ symmetry line, and $\left(+-\right)$ stretches indicated by green dashed lines are located in regions with $\mu \ne \nu$ away from the symmetry line. The connections between (0) and $\left(+-\right)$ stretches are drawn with thin black lines. $\left(+-\right)$ stretches can be mirrored at the $\mu =\nu$ symmetry line (rule R-1), traversed backward (rule R-2), and the orders of the (0) or $\left(+-\right)$ stretches can be interchanged (rules R-3 and R-4).

Figure 6

(Color online) Distribution histogram of the classical actions for the 16 trajectories of the near–action-degenerate periodic-orbit bunch shown in Figs. 3, 4. The small differences between the actions of individual orbits are only visible in the enlarged inset.

Figure 7

(Color online) Action difference $\Delta s$ between two orbits consisting of a (0) stretch with $n$ zero symbols and a $\left(+-\right)$ stretch with $L-n-1$ minus symbols. Evidently, $\Delta s$ decreases exponentially with the length of the (0) stretch.

Figure 8

(Color online) Number of periodic orbits with classical action $s_{\text{po}}<s$ for all individual orbits (dashed green line), only one representative of a periodic-orbit bunch (dotted blue line), and one representative of a periodic-orbit bunch weighted with the size of the bunch (solid red line, nearly indistinguishable from the dashed green line). All curves show an exponential increase, however, the number of periodic-orbit bunches grows with a significantly lower rate than the number of all orbits.

Figure 9

(Color online) Resonances with (a) even and (b) odd $z$ parity of the diamagnetic hydrogen atom at scaled energy $\tilde{E}=0.5$. In most cases especially the real parts of the semiclassical resonances obtained with the reduced data set (red crosses) agree very well with the semiclassical resonances obtained with the complete periodic-orbit set (blue squares). For completeness and comparison the numerically exact quantum resonances are shown as black dots.