Dynamical and Quenched Random Matrices and Homolumo Gap

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.