Periodic orbit theory and spectral rigidity in pseudointegrable systems

We calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers $g$ that arise from rectangular systems with one or two salient corners. From the periodic orbits, we calculate the spectral rigidity ${\Delta }_3\left(L\right)$ using semiclassical quantum mechanics with $L$ reaching up to quite large values. We find that the diagonal approximation is applicable when averaging over a suitable energy interval. Comparing systems of various shapes, we find that our results agree well with ${\Delta }_3$ calculated directly from the eigenvalues by spectral statistics. Therefore, additional terms such as, e.g., diffraction terms seem to be small in the case of the systems investigated in this work. By reducing the size of the corners, the spectral statistics of our pseudointegrable systems approaches that of an integrable system, whereas very large differences between integrable and pseudointegrable systems occur when the salient corners are large. Both types of behavior can be well understood by the properties of the periodic orbits in the system.

Figure 1

Shapes of the pseudointegrable systems considered in this paper: (a) the one-step system $(X_1,Y_1;X_2,Y_2)$ with genus number $g=2$ and (b) the two-step system $(X_1,Y_1;X_2,Y_2;X_3,Y_3)$ with $g=3$. In (a), the beam splitting of two initially neighboring trajectories at a salient corner is demonstrated and in (b) trajectories of three ”neutral” orbits (between two parallel walls) are indicated by arrows. For the lengths $X_i,Y_i$ considered in this paper, see Table .

Figure 2

Test of the sum rule [see Eq. (6)] for the geometries of Fig. 1 with different values of $X_i,Y_j$. $S\left(\ell \right)$ is plotted versus $\ell$ (giving straight lines) and $b_0$ is calculated from the slopes. The different symbols refer to the rectangular system (triangular symbols) and to pseudointegrable systems of $g=2$ (circles) and $g=3$ (squares). Open and closed symbols refer to different step sizes (for details of the systems, see Table ). For the rectangle, we find $b_0=\frac{1}{4}$ and for the pseudointegrable systems, $b_0$ is slightly increasing with $g$ and with the step sizes.

Figure 3

Normalized average area $⟨a\left(\ell \right)⟩$ in phase space for the periodic orbits with lengths smaller than $\ell$ for the same systems (and same symbols) as in Fig. 2. It can be seen that systems of the same genus number $g$ form groups of similar $⟨a\left(\ell \right)⟩$. For large values of $\ell$, the areas are saturating.

Figure 4

The proliferation rate $N\left(\ell \right)$ is plotted versus $\ell$ for five different systems and two types of calculations. The lines are calculated according to Eq. (4), using the values of $b_0$ and $⟨a\left(\ell \right)⟩$ from Figs. 2, 3, respectively. The symbols (same symbols as in Fig. 2) show the numerical data, gained by a simple summation of the different periodic orbits (including repetitions). Both curves agree very well.

Figure 5

$I\left(\ell \right)$ for a rectangle (upper three curves) and a two-step system (lower three curves) (for details of the systems, see Table ). In both cases, the diagonal approximation $I\left(\ell \right)$ is close to a straight line (solid line), whereas the full sum oscillates around it (dotted and dashed lines). The oscillations are significantly reduced by averaging over more energy values. The sums are carried out over the first 3000 orbits, including repetitions.

Figure 6

${\Delta }_3\left(L\right)$ is calculated by the diagonal approximation (open symbols) and by the full sum (solid symbols) and plotted vs $L$ for different systems of genus number $g=1$ (squares), $g=2$ (circles), and $g=3$ (triangles) (for details, see Table ). The theoretical behavior for integrable and chaotic systems is indicated by the upper and the lower lines, respectively (dotted lines without symbols).

Figure 7

${\Delta }_3\left(L\right)$ from the periodic orbit theory in the diagonal approximation (open symbols) is compared to ${\Delta }_3$ calculated from the eigenvalues under Neumann boundary conditions obtained by the Lanczos algorithm (solid symbols) for different systems. For the rectangle, ${\Delta }_3$ as calculated from the exact eigenvalues for a continuous rectangluar system is shown as well (solid diamonds). Different symbols represent the genus numbers $g=1$ (squares), $g=2$ (circles), and $g=3$ (triangles). The theoretical behavior for integrable and chaotic systems is indicated by the upper and the lower line, respectively (dotted lines without symbols).