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.