Statistical Theory of the Energy Levels of Complex Systems. IV

Abstract

In previous papers of this series, a theory was constructed for the description of the statistical properties of the eigenvalues of a random matrix of high order. This paper deduces from the same theoretical model the statistical behavior to be expected for a finite stretch of observed eigenvalues chosen out of a much longer stretch of unobserved ones. In comparing the model with experimental data, we have always to deal with such a finite stretch of observed levels. Three ``statistics'' are investigated; these are quantities which can be computed directly from the observed data, and for which the expectation values and statistical variances can be calculated from the theory. One statistic gives only a precise way of measuring the average level spacing. One provides a test of the model by measuring the degree of long-range order in the level series. The third provides an independent test of the model by measuring the degree of short-range order. In Sec. V these methods are applied to a preliminary analysis of some experimental data on neutron capture levels in heavy nuclei. The results are inconclusive. Discrepancies between theory and observation are large, but the discrepancies might be produced either by incompleteness of the data or by incorrectness of the theoretical model.