Pair correlation of four-dimensional flat tori
Berry and Tabor (1977) conjectured that the local spacings between the eigenvalues of the quantization of the Hamiltonian for generic, completely integrable systems follow the distribution of local spacings between random numbers. \textit{P. Sarnak} [CMS Conf. Proc. 21, 181-203 (1997; Zbl 0911.11032)] showed that the pair correlations of eigenvalues on almost all two-dimensional flat tori have this behaviour. \textit{J. Vanderkam} [ibid. 97, 397-412 (1999; Zbl 0965.11016)] extended this result to almost all \(n\)th degree forms in \(n\) variables in a measure theoretic sense. In this paper the author considers a set of measure zero among all four variables homogeneous quartic forms. These correspond to the eigenvalues of flat tori as follows: Let \(P\) be a positive definite homogeneous polynomial of degree four in four variables. Let \(0= \lambda_0< \lambda_1\leq \lambda_2\leq\dots\) be a discrete ordered set of values taken by \(P\) at integers. Let \(C_p\) be the volume of the region \(\{P (\overline{x})< 1\}/ (\overline{x}\sim -\overline{x})\). The number of \(\lambda\)'s less than \(N\) counted with multiplicity is essentially \(C_p N\). Consider the distributions of local spacings \(\lambda_j- \lambda_k\) and the consecutive spacings \(\lambda_{j+1}- \lambda_j\). Let \[ R_p(a,b,N)= \frac {|\{\lambda_j, \lambda_k< n\mid j\neq k, a\leq \lambda_j- \lambda_k\leq b\}|} {N} \] be their pair correlations. Then for almost all \(P\), which are squares of four variable quadratic forms \(R_p(a,b,N)\to C_p^2 (b-a)\) for all \(b> a\). This implies that almost all four-dimensional flat tori have spectra whose desymmetrized pair correlation satisfies the Berry-Tabor conjecture and there is a set of second Baire category among the four-dimensional flat tori for which the consecutive spacing distribution does not converge. The method used in this paper depends heavily on the factorization that is unique to four-dimensional flat tori, so they do not generalize immediately to higher dimensions.
scientific article; zbMATH DE number 1425201