  We consider a rather general type of matrix model, where the matrix M
represents a Hamiltonian of the interaction of a bosonic system with a single
fermion. The fluctuations of the matrix are partly given by some fundamental
randomness and partly dynamically, even quantum mechanically. We then study the
homolumo-gap effect, which means that we study how the level density for the
single-fermion Hamiltonian matrix M gets attenuated near the Fermi surface. In
the case of the quenched randomness (the fundamental one) dominating the
quantum mechanical one we show that in the first approximation the homolumo gap
is characterized by the absence of single-fermion levels between two steep gap
boundaries. The filled and empty level densities are in this first
approximation just pushed, each to its side. In the next approximation these
steep drops in the spectral density are smeared out to have an error-function
shape. The studied model could be considered as a first step towards the more
general case of considering a whole field of matrices - defined say on some
phase space - rather than a single matrix.
