The 2-point correlation form factor, $K_2(\tau)$, for small values of $\tau$
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 $\pi/n$ 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
$K_2(0)=(n+\epsilon(n))/(3(n-2))$ where $\epsilon(n)=0$ for odd $n$,
$\epsilon(n)=2$ for even $n$ not divisible by 3, and $\epsilon(n)=6$ 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, $K_2(0)=1-3\bar{\alpha}+4\bar{\alpha}^2$ where $\bar{\alpha}$ is the
fractional part of the flux through the rectangle when $0\le \bar{\alpha}\le
1/2$ and it is symmetric with respect to the line $\bar{\alpha}=1/2$ when $1/2
\le \bar{\alpha}\le 1$. 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.Comment: 61 pages, 13 figures. Submitted to Communications in Mathematical
Physics, 200