98-61 Jens Marklof

Spectral form factors of rectangle billiards (337K, LATeX 2e) Feb 16, 98

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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.

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