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

Abstract

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.

61 pages, 13 figures. Submitted to Communications in Mathematical Physics, 2000

References in corpus (1)

• Singular statistics

Cited by in corpus (16)

• Trace formula for dielectric cavities II: Regular, pseudo-integrable, and chaotic examples
• Intermediate statistics in quantum maps
• Spectral properties of quantized barrier billiards
• Spectral Statistics of Rectangular Billiards with Localized Perturbations
• Spectral statistics of a pseudo-integrable map: the general case
• Point perturbations of circle billiards
• Low rank perturbations and the spectral statistics of pseudointegrable billiards
• Evanescent wave approach to diffractive phenomena in convex billiards with corners
• Quantum Dynamical Tunneling Breaks Classical Conserved Quantities
• Barrier billiard and random matrices
• Spectral Properties of Quantum Circulant Graphs
• Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph
• Localized Perturbations of Integrable Systems
• Spectral form factors and dynamical localization
• Periodic orbits and semiclassical form factor in barrier billiards
• Formation of superscar waves in plane polygonal billiards