Correlations of chaotic eigenfunctions: a semiclassical analysis

We derive a semiclassical expression for an energy smoothed autocorrelation function defined on a group of eigenstates of the Schr\"odinger equation. The system we considered is an energy-conserved Hamiltonian system possessing time-invariant symmetry. The energy smoothed autocorrelation function is expressed as a sum of three terms. The first one is analogous to Berry's conjecture, which is a Bessel function of the zeroth order. The second and the third terms are trace formulae made from special trajectories. The second term is found to be direction dependent in the case of spacing averaging, which agrees qualitatively with previous numerical observations in high-lying eigenstates of a chaotic billiard.Comment: Revtex, 13 pages, 1 postscript figur

Multiple Recurrent De Novo CNVs, Including Duplications of the 7q11.23 Williams Syndrome Region, Are Strongly Associated with Autism

SummaryWe have undertaken a genome-wide analysis of rare copy-number variation (CNV) in 1124 autism spectrum disorder (ASD) families, each comprised of a single proband, unaffected parents, and, in most kindreds, an unaffected sibling. We find significant association of ASD with de novo duplications of 7q11.23, where the reciprocal deletion causes Williams-Beuren syndrome, characterized by a highly social personality. We identify rare recurrent de novo CNVs at five additional regions, including 16p13.2 (encompassing genes USP7 and C16orf72) and Cadherin 13, and implement a rigorous approach to evaluating the statistical significance of these observations. Overall, large de novo CNVs, particularly those encompassing multiple genes, confer substantial risks (OR = 5.6; CI = 2.6–12.0, p = 2.4 × 10-7). We estimate there are 130–234 ASD-related CNV regions in the human genome and present compelling evidence, based on cumulative data, for association of rare de novo events at 7q11.23, 15q11.2-13.1, 16p11.2, and Neurexin 1