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