Pseudointegrable systems with growing surface roughness

Stefanie Russ, Yuriy Hlushchuk

Institut für Theoretische Physik III
Universität Giessen, Germany

We study numerically the level statistics of pseudointegrable billards with different degrees of surface roughness, expressed by the different genus numbers $g$ and calculate the eigenvalues and the eigenfunctions under Dirichlet and Neumann boundary conditions.

We study the distance distribution $P(s)$ of the eigenvalues and related quantities such as half of the second moment $I_0=(1/2)\int s^2 P(s) ds$, the spectral rigidity $\Delta_3$ and the number variance $\Sigma^2$. The considered systems possess genus numbers between $g=2$ and $g\approx 700$. Their shapes range from very simple two-step systems to systems with many small teeth at the boundary and fractal drums with complicated scaling surface roughness. The eigenfunctions are analyzed according to their localization volume $V_{\rm {loc}}$ (inverse participation ratio), their amplitude distribution $P(\psi)$ and the momentum space.

For lower energies we can distinguish several energy windows with different behavior. In some windows, the values of $I_0$ come close to the value of the Poisson distribution $I_0=1$. The eigenfunctions in these regimes are either localized (with $V_{\rm {loc}}\le 0.2\,V_0$, where $V_0$ is the surface of the system) or relatively regular functions with $V_{\rm {loc}} \approx 4/9 V_0$, the value of a sine or cosine function. In other energy windows, the values of $I_0$ come closer to the Wigner limit of $I_0\approx 0.637$ and the eigenfunctions approach chaotic functions with a Gaussian distribution $P(\psi)$.

Except for some unusual nearly-symmetric system shapes, we find an asymptotic energy regime above some non-universal limiting energy value. In the asymptotic regime, the genus number $g$ seems to be the determining parameter that governs the level statistics. With increasing $g$, the behavior of the systems in the asymptotic regime changes from Poisson-like to Wigner-like behavior.