Jens Marklof
Spectral form factors of rectangle billiards
(337K, LATeX 2e)
ABSTRACT. The Berry-Tabor conjecture asserts that local statistical measures of the
eigenvalues $\lambda_j$ of a ``generic'' integrable quantum system
coincide with those of a Poisson process. We prove that, in the case of a
rectangle billiard with random ratio of sides, the sum $\sum_{j\leq N}
\exp(2\pi i\lambda_j\tau)$ behaves for $\tau$ random and $N$ large like a
random walk in the complex plane with a non-Gaussian limit distribution.
The expectation value of the distribution is zero; its variance, which is
essentially the average pair correlation function, is one, in accordance
with the Berry-Tabor conjecture, but all higher moments ($\geq 4$)
diverge. The proof of the existence of the limit distribution uses the
mixing property of a dynamical system defined on a product of hyperbolic
surfaces. The Berry-Tabor conjecture and the existence of the the limit
distribution for a fixed generic rectangle are related to an
equidistribution conjecture for long horocycles on this product space.