Analysis of spectral densities using modified moments

Abstract

Modified moments provide coefficients in orthogonal polynomial expansions for spectral densities. Using these expansions as a starting point, we develop a number of methods which can be used in the analysis of density functions. Expansions for averages over densities are described. These expansions, when combined with nonlinear convergence acceleration methods based on the Pad\'e approximant, give apparently exponentially convergent results. By exploiting connections between the orthogonal polynomial expansion, Fourier series, and power series, we show how to obtain an accurate picture of the density itself. A rational approximation is described which gives very accurate results near the ends of the interval. Procedures are given for determining the number, types, and locations of singularities in a density from its modified moments. By an analysis of the asymptotic form of the modified moments, we show how this information about singularities can be incorporated into an expansion for the density using the "moment-singularity" method. We illustrate these methods by applications to harmonic solid models. We obtain extremely accurate results for averages and we obtain accurate representations for spectral densities which faithfully reproduce the Van Hove singularities.