Distribution of the lowest eigenvalue for a class of random matrices

Abstract

The random matrix theory is applied to the ensemble of partially diagonalized shell-model Hamiltonian matrices. Using the analogy with the distribution of the poles of the unitary collision matrix, an expression for the joint probability density function of the eigenvalues is derived for the present ensemble. An approximation is used to work out a closed form expression for the probability density function of the lowest eigenvalue. Two simple examples are considered to see what this probability density function looks like.