Exact Results for the Tavis-Cummings and Hückel Hamiltonians with Diagonal Disorder

We present an exact method to calculate the electronic states of one electron Hamiltonians with diagonal disorder. We show that in cases where the disorder has a Cauchy distribution, the disorder averaged one particle Green's function can be calculated directly, using a deterministic, complex (non-Hermitian) Hamiltonian. For this we use the supersymmetric method which has already been used in problems of solid state physics. Using the method we find exact solution for the case of N molecules with site disorder, confined to a microcavity, for any value of N. Our analysis shows that the width of the polaritonic states as a function of N depends on the nature of disorder, and hence it can be used to probe the way molecular energy levels are distributed. We also show how one can find exact results for Hückel type Hamiltonians with on-site Cauchy disorder and demonstrate its use.