The effect of disorder on the spectrum of a Hermitian matrix

It is well known that the average distribution of eigenvalues of a matrix, whose elements have a Gaussian distribution, may be determined. Here it is shown that the sum of such a matrix with a non-fluctuating matrix can also be resolved, in as much as the problem can be reduced to the solution of a self-consistent Dyson equation, a non-linear equation well known in many-body theory.