Eigenvalues of quantizations of completely integrable Hamiltonians.

Let $\alpha$ be an irrational of ``diophantine type'', for example $\alpha = \sqrt{2}$ . For N large, consider the N numbers in [0,1), $n^2\alpha \ (\text{mod } 1 )$ , $n=1,2,\dots,N$ . Order these as $0 \le x_1 \le x_2 \le \dots \le x_N < 1$ . Do the normalized consecutive spacings

$\begin{displaymath}\Delta _j = N(x_{j+1} - x_j),\ \ j=1,\dots,N-1 \end{displaymath}$

behave like spacings between random numbers? So for example, does the distribution of the consecutive spacings tend to the density $e^{-x}\,dx$ ? This problem arises in connection with the spacing distribution for the eigenvalues of quantizations of completely integrable Hamiltonians, and was raised by physicists.

ROBERT LIPSHITZ'S INVESTIGATIONS