Pair correlation densities of inhomogeneous quadratic forms. II. (English) Zbl 1136.11325
Summary: Denote by \(\|\cdot\|\) the Euclidean norm in \(\mathbb {R}^ k\). We prove that the local pair correlation density of the sequence \(\|\mathbf{m}- \alpha\|^ k,\text\textbf{m}\in \mathbb {Z}^ k\), is that of a Poisson process, under Diophantine conditions on the fixed vector \(\alpha\in \mathbb {R}^ k\) in dimension two, vectors \(\alpha\) of any Diophantine type are admissible; in higher dimensions \((k>2)\), Poisson statistics are observed only for Diophantine vectors of type \(\kappa<(k-1)/(k-2)\). Our findings support a conjecture of M. V. Berry and M. Tabor [Proc. R. Soc. Lond., Ser. A 356, 375–394 (1977; Zbl 1119.81395)] on the Poisson nature of spectral correlations in quantized integrable systems.
MSC:
| 11P21 | Lattice points in specified regions |
| 11F27 | Theta series; Weil representation; theta correspondences |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Citations:
Zbl 1119.81395References:
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