The spectral shape of nonlinear chains: validity of perturbative and moment approaches

We test the range of validity of the perturbative approach for calculating the spectral shape of a one-dimensional Lennard-Jones chain. We reconstruct the spectral shape from its frequency moments through the use of a continued fraction representation with truncation. The quantum moments for the Lennard-Jones potential are calculated by means of the effective potential method. The classical frequency moments are calculated exactly for this potential and for the approximation to it obtained by expanding it in powers of the deviations from the position of its minimum through terms of fourth order. The comparison with molecular dynamics results for the same potentials confirms the validity of the continued fraction approach, and shows that the perturbative approach is valid only at very low temperatures.