Transfer matrices, non-Hermitian Hamiltonians and Resolvents: some spectral identities

Abstract

I consider the N-step transfer matrix T for a general block Hamiltonian, with eigenvalue equation L_n ψ_{n+1} + H_n ψ_n + L_{n-1}^\dagger ψ_{n-1} = E ψ_n where H_n and L_n 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 ψ_0 =ψ_{N+1} =0. I then introduce the related Hamiltonian for the case ψ_0 = z^{-1} ψ_N and ψ_{N+1} = z ψ_1, 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=e^{iϕ}. The results are then extended to T^\dagger T by showing that it is itself a transfer matrix. Besides their own 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. P.A.C.S.: 02.10.Sp (theory of matrices), 05.60 (theory of quantum transport), 71.23 (Anderson model), 72.17.Rn (Quantum localization)