The absence of the Poisson distribution of spacings between quasi- energies in the quantum kicked-rotator model
Let (i) T be a skew-rotation of the 2-dimensional torus \(M=S\times S\), \((S=R mod Z)\) defined by T(x,y)\(\equiv (x+\alpha,y+x)\), where \(\alpha\) is irrational; (ii) \(\xi^{(\epsilon)}(x,y)= the\) number of k in \(\{0,1,...,N_{\epsilon}\}\) for which \(T^ k(x,y)\in \Pi_{\epsilon}\equiv \{(x,y):\) \(| y| \leq \epsilon /2\}\), here \(N_{\epsilon}\) is the greatest integer not exceeding \(\epsilon^{-1}\). It was conjectured that the limiting distribution of \(\xi^{(\epsilon)}\) as \(\epsilon\) \(\downarrow 0\) is Poisson (with parameter 1). Indeed the first 4 moments \(M_ r[\xi^{(\epsilon)}]=\iint [\xi^{(\epsilon)}(x,y)]^ r dx dy\) tend to the moments of the Poisson distribution. But this paper shows that for typical \(\alpha\) the limit distribution of \(\xi^{(\epsilon)}\) cannot be Poisson (Theorem 2) and that for \(r\geq 27\), \(\lim_{\epsilon \downarrow 0}\int M_ r[\xi^{(\epsilon)}]d\alpha =\infty\) (Theorem 1).
scientific article; zbMATH DE number 4082609