Homolumo Gap from Dynamical Energy Levels

We introduce a dynamical matrix model where the matrix $X$ is interpreted as a Hamiltonian representing interaction of a bosonic system with a single fermion. We show how a system of second-quantized fermions influences the ground state of the whole system by producing a gap between the highest occupied eigenvalue and the lowest unoccupied eigenvalue. We describe the development of the gap in both, strong and weak coupling regime, while for the intermediate coupling strength we expect formation of homolumo "kinks".