Anomalies in the 1D Anderson model: Beyond the band-centre and band-edge cases

We consider the one-dimensional Anderson model with weak disorder. Using the Hamiltonian map approach, we analyse the validity of the random-phase approximation for resonant values of the energy, E=2cos(πr), with r a rational number. We expand the invariant measure of the phase variable in powers of the disorder strength and we show that, contrary to what happens at the centre and at the edges of the band, for all other resonant energies the leading term of the invariant measure is uniform. When higher-order terms are taken into account, a modulation of the invariant measure appears for all resonant values of the energy. This implies that, when the localisation length is computed within the second-order approximation in the disorder strength, the Thouless formula is valid everywhere except at the band centre and at the band edges.