Periodic orbits contribution to the 2-point correlation form factor for pseudo-integrable systems

Abstract

The 2-point correlation form factor, $K_2(τ)$, for small values of $τ$ is computed analytically for typical examples of pseudo-integrable systems. This is done by explicit calculation of periodic orbit contributions in the diagonal approximation. The following cases are considered: (i) plane billiards in the form of right triangles with one angle $π/n$ and (ii) rectangular billiards with the Aharonov-Bohm flux line. In the first model, using the properties of the Veech structure, it is shown that $K_2(0)=(n+ε(n))/(3(n-2))$ where $ε(n)=0$ for odd $n$, $ε(n)=2$ for even $n$ not divisible by 3, and $ε(n)=6$ for even $n$ divisible by 3. For completeness we also recall informally the main features of the Veech construction. In the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle. When these ratios are non-commensurable irrational numbers, $K_2(0)=1-3\barα+4\barα^2$ where $\barα$ is the fractional part of the flux through the rectangle when $0\le \barα\le 1/2$ and it is symmetric with respect to the line $\barα=1/2$ when $1/2 \le \barα\le 1$. The comparison of these results with numerical calculations of the form factor is discussed in detail. The above values of $K_2(0)$ differ from all known examples of spectral statistics, thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo-integrable systems.