arXiv subjects
Publications and source records attributed to Debabrata Biswas.
We review some properties of periodic orbit families in polygonal billiards and discuss in particular a sum rule that they obey. In addition, we provide algorithms to determine periodic orbit families and present numerical results that shed new light on the proliferation law and its variation with the genus of the invariant surface. Finally, we deal with correlations in the length spectrum and find that long orbits display Poisson fluctuations.
We study ray dynamics in a square billiard that allows mode conversion and is parametrised by $κ$, the ratio of the velocities of the two modes. At $κ\rightarrow 1^+$, conversion occurs at every reflection and periodic orbits proliferate exponentially. As $κ$ increases beyond $\sqrt{2}$, the collection of daughter rays explore only three momentum directions and mode conversion is progressively inhibited. We provide an algorithm for determining periodic orbits when $κ> \sqrt{2}$ and show numerically that exponential proliferation persists around $κ\simeq \sqrt{2}$ but as $κ$ increases, a crossover to sub-exponential behaviour occurs for short periods. We discuss these results in the light of conservation laws.
We demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than $l$ increases as $πb_0l^2/\langle a(l) \rangle$, where $b_0$ is a constant and $\langle a(l) \rangle$ is the average area occupied by these families. We also find that $\langle a(l) \rangle$ increases with $l$ before saturating. Finally, we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies.