Saturation of the width of the strength function ∗

The strength function of a single state |d〉 is studied using the deformed Gaussian orthogonal ensemble. In particular we study the dependence of the spreading width of |d〉 on the degree of mixing. The mixing of a single state with a background of complicated states is important in the description of a variety of phenomena in nuclear physics such as isobaric analogue resonances, giant dipole resonances and the decay out superdeformed rotational bands. Such can mixing be conveniently described by the strength function [1] and it is interesting to study the generic features of this object. Previous studies in this vein have investigated a single state coupled to a background generated by a two-dimensional anharmonic oscillator [2] and the spreading of a shell model basis state over the shell model eigenstates due to the the residual interaction [3]. Here we use random matrix theory [4]. We write the Hamiltonian, H, as the sum of two terms, H = H0 + V. (1) The eigenstates, |n〉, and eigenvalues, En, of H satisfy H|n〉 = En|n〉, n = 1, ...N + 1, (2) whilst for H0 we have H0|k〉 = Ek|k〉, k = 1, ...N, H0|d〉 = Ed|d〉, d = N + 1. (3) The strength function is then defined as