# Strength distributions and statistical spectroscopy. I. General theory

#### Abstract

The strength distribution for an arbitrary excitation is given in terms of a double expansion, and its sum rules by single expansions, in polynomials defined by the initial and final energy spectra. In model spaces which are not too large, a rapid convergence, to within fluctuations, is assured by the action of a central limit theorem, as is shown in particular by considering the response of the system under infinitesimal deformations of the Hamiltonian. When larger spaces are decomposed into subspaces defined by a partitioning of the single-particle space a similar convergence results. At the same time, close contact is made with, and important corrections are found to, intuitive procedures which are often used for approximating strength distributions. The general features of the distribution are often easily understood in termsof a simple geometry made effective in the model space by the central limit theorem, and further features by exploiting the connection of this geometry to the unitary group of transformations in the single-particle space. Extensions are given for multipole strengths and sum rules, appropriate when the angular momenta (and isospins) are specified for the states involved in the transitions. Measures for the RMS fluctuations in the sum-rule quantities, and correlations between them, are given by combining the low-order-polynomial (statistically smoothed) strengths with an assumed Porter-Thomas distribution for the (high-order) strength fluctuations. © 1977.

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