Energy--Level Statistics of Model Quantum Systems: Universality and Scaling in a Lattice--Point Problem

Abstract

We investigate the statistics of the number $N(R,S)$ of lattice points, $n\in {\mathbb{Z}}^2$, in a ``random'' annular domain $\Pi (R,w)=(R+w)A\setminus RA$, where $R,w>0$. Here $A$ is a fixed convex set with smooth boundary and $w$ is chosen so that the area of $\Pi (R,w)$ is $S$. The randomness comes from $R$ being taken as random ( with a smooth denisity ) in some interval $[c_{1}T,c_{2}T]$, $c_2>c_1>0$. We find that in the limit $T\to \infty$ the variance and distribution of $\De N=N(R;S)-S$ depends strongly on how $S$ grows with $T$. There is a saturation regime $S/T\to \infty$, as $T\to \infty$ in which the fluctuations in $\Delta N$ coming from the two boundaries of $\Pi$, are independent. Then there is a scaling regime, $S/T\to z$, $0<z<\infty$ in which the distribution depends on $z$ in an almost periodic way going to a Gaussian as $z\to 0$. The variance in this limit approaches $z$ for ``generic'' $A$ but can be larger for ``degenerate'' cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.

arXiv comment48 pages, IASSNS-HEP 93/14

Cited by in corpus (7)

• Level Spacings for Integrable Quantum Maps in Genus Zero
• Quantum jumps of saturation level rigidity and anomalous oscillations of level number variance in the semiclassical spectrum of a modified Kepler problem
• On the distribution of lattice points in thin annuli
• Vortices and Magnetization in Kac's Model
• Statistics of lattice points in thin annuli for generic lattices
• The distribution of lattice points in elliptic annuli
• Distribution of Energy Levels of a Quantum Free Particle on a Surface of Revolution