SINGULARITIES IN THE SPECTRA OF RANDOM MATRICES

We consider singularities of the set of energy levels E(n)(X) of a qua ntum Hamiltonian, obtained by varying a set of d parameters X = (X(1), ..,X(d)). Singularities such as minima, degeneracies, branch points, a nd avoided crossings can play an important role in physical applicatio ns. We discuss a general method for counting these singularities, and apply it to a random matrix model for the parameter dependence of ener gy levels. We also show how the density of avoided crossing singularit ies is related to a non-analyticity of a correlation function describi ng the energy levels. (C) 1996 American Institute of Physics.