Marginal distribution of the S-matrix elements for Dyson's measure and some applications

In recent attempts to construct a statistical theory of nuclear reactions by doing statistics directly on the S-matrix elements, Dyson's measure, which remains invariant under an automorphism that maps the space of unitary and symmetric matrices into itself, is of fundamental importance. The authors study some of the marginal distributions of the individual S-matrix elements, or of groups of them, that arise from Dyson's measure. To understand the problem better, a similar discussion is first carried out for Haar's measure of unitary matrices which do not have the restriction of symmetry, and some of the effects of this restriction are thus exhibited. Some applications of these mathematical results to reaction theory are discussed.