Transfer matrices, non-Hermitian Hamiltonians and resolvents: Some spectral identities

I consider the N-step transfer matrix T for a general block Hamiltonian, with eigenvalue equation Lnψn+1 + Hnψn + L†n-1 ψn-1 = Eψn where Hn and Ln are matrices, and provide its explicit representation in terms of blocks of the resolvent of the Hamiltonian matrix for the system of length N with boundary conditions ψ0 = ψN+1 = 0. I then introduce the related Hamiltonian for the case ψ0 = z-1ψN and ψN+1 = zψ1, and provide an exact relation between the trace of its resolvent and Tr(T - z)-1, together with an identity of Thouless type connecting Tr (log|T|) with the Hamiltonian eigenvalues for z = eiφ. The results are then extended to T†T by showing that it is itself a transfer matrix. Besides being of mathematical interest, the identities should be useful for an analytical approach in the study of spectral properties of a physically relevant class of transfer matrices.