Application of the trace formula in pseudointegrable systems

We apply periodic-orbit theory to calculate the integrated density of states $N\left(k\right)$ of the quantum mechanical eigenvalues from the periodic orbits of pseudointegrable polygon and barrier billiards. We show that the results agree so well with the density of states obtained from numerical solutions of the Schrödinger equation that about the first 100 eigenvalues can be obtained directly from the periodic-orbit calculations with good accuracy.

Figure 1

Shapes of the pseudointegrable billiards considered in this work. (a), (b) L-shaped billiards (polygon billiards of genus number $G=2$), (c) a polygon billiard of $G=3$, and (d) the barrier billiard $(G=2)$, which is calculated for different heights $h$ of the barrier. One periodic orbit for each geometry is also shown (dashed lines). In (b) the different segments are shown that can be transversed by the periodic orbits.

Figure 2

(a), (b) Density of states $N^{\mathrm{PO}}\left(k\right)$ calculated from the periodic orbits (solid lines) and $N^{\mathrm{EV}}\left(k\right)$ calculated from the exact eigenvalues (circles) of a rectangular system of side lengths $L_x=101$ and $L_y=198$. In (b), the upper curve compares the steps in $N^{\mathrm{PO}}\left(k\right)$ to the eigenvalues. In the lower curves (shifted down by a value of 5), the areas of the orbits are disturbed by errors of up to 0.1% and it can be seen that the steps in $N^{\mathrm{PO}}\left(k\right)$ (dotted line) are strongly disturbed already by these small error bars.

Figure 3

(a), (b) Density of states $N^{\mathrm{PO}}\left(k\right)$ calculated from the periodic orbits (solid lines) and $N^{\mathrm{EV},\mathrm{L}}\left(k\right)$ from the Lanczos eigenvalues for the L-shaped systems of Figs. 1a (circles) and 1b (squares) and the system of genus number $G=3$ of Fig. 1c (diamonds). In (b), the fitted step function of $N_{\mathrm{fit}}^{\mathrm{PO}}\left(k\right)$ is compared to the Lanczos eigenvalues. For a better overview the data of the geometries of Figs. 1a, 1b have been shifted upward by values of 150 in (a) and by 25 in (b).

Figure 4

(a), (b) Density of states $N^{\mathrm{PO}}\left(k\right)$ calculated from the periodic orbits (solid lines) and $N^{\mathrm{EV},\mathrm{L}}\left(k\right)$ from the Lanczos eigenvalues for the barrier billiards with barrier heights $h=10$ (diamonds), 50 (squares), and 100 (circles). In (b), the fitted step function of $N_{\mathrm{fit}}^{\mathrm{PO}}\left(k\right)$ is compared to the Lanczos eigenvalues. For a better overview the data in both (a) and (b) have been shifted upward by values of 5 $(h=50)$ and 10 $(h=10)$.