Triangles with sides given by consecutive integers ($N$, $N+1$, $N+2$) are fully irrational (all angles irrational with $\pi$) if $3<N<\infty$. Rational approximations to their angles and the Hurwitz theorem in number theory are used to define a parameter $h$ that quantifies the irrationality of each triangle. The energy level statistics [spacing distribution $p\left(s\right)$ and spectral rigidity ${\Delta }_3\left(L\right)$] of quantum billiards from this one-parameter family of triangles are investigated. The behavior of $h$ with varying $N$ and the numerically calculated level dynamics are found to be closely related: $h$ exhibits a local maximum at $N=10$, around which agreement with Gaussian orthogonal ensemble (GOE) spectral fluctuations is observed. As $N$ is increased, $h$ decreases and the statistics depart from GOE. Structures appear in $p\left(s\right)$ for $N>120$ and eventually the occurrence of gaps in the distribution for $N\sim 180$ define the onset of a long crossover towards the sequence observed in the integrable limit of the equilateral triangle $(N\to \infty )$.

• Received 15 June 2007

DOI:https://doi.org/10.1103/PhysRevE.77.036201