Statistical Description of the Complex-Boundary-Value Problem. II. Distribution of the Parameters of the Collision Matrix

Abstract

A detailed statistical study is made of the parameters of the statistical collision matrix using the $N$-dimensional random complex orthogonal matrix. It is shown that, even without a complete knowledge of weight function which has to be introduced for the convergence of the normalization integral, certain relations between the average values of the parameters of the statistical collision matrix can be obtained and a statement can be made that the channel correlations of the parameters are always positive. A suitable form of the weight function is guessed, and the distributions of the parameters are also given. It is shown that under certain conditions the distribution of the parameters is close to the Porter-Thomas distribution except for small values. The resonance correlations of the parameters are also studied. Excellent agreement has been obtained between the values predicted by the present theoretical formulation and those obtained by a numerical calculation using the parameters of the real-boundary-value problem and a certain transformation matrix.