The absence of the Poisson distribution of spacings between quasi- energies in the quantum kicked-rotator model
type: Item
instance of: scholarly article
title: The absence of the Poisson distribution of spacings between quasi- energies in the quantum kicked-rotator model
zbMATH Open document ID: 0662.60031
DOI: 10.1016/S0167-2789(98)90024-0
published in: Physica D
publication date: 1988-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 4082609
zbMATH Keywords: quantum rotator
zbMATH Keywords: skew-rotation on torus
zbMATH Keywords: Poisson distribution
zbMATH Keywords: limit distribution
author: Ya. G. Sinai
MaRDI profile type: Publication
cites work: Q4291790
cites work: Evidence for the Poisson distribution for quasi-energies in the quantum kicked-rotator model
full work available at URL: s0167-2789(98)90024-0
OpenAlex ID: W2009803877
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description: scientific article; zbMATH DE number 4082609