Numerical Approximation of Singular Spectral Functions Arising from the Fourier-Jacobi Problem on a Half Line with Continuous Spectra

Abstract

In this paper we show that the classical Fourier-Jacobi inversion integral is associated with a second order Sturm-Liouville problem on the half line (0, ℞) with both endpoints singular. The associated eigenfunction expansion is written with one spectral function (the spectrum being simple), and Subroutine SLEDGE is used to compute both discrete eigenvalues and the singular spectral density function over the range of continuous spectrum. An independent check on accuracy for the singular spectral density function is given by numerical computation of a closed form formula involving an integral of the gamma function.