Efficient calculation of spectral density functions for specific classes of singular Sturm–Liouville problems

摘要

New families of approximations to Sturm–Liouville spectral density functions are derived for cases where the potential function has one of several specific forms. This particular form dictates the type of expansion functions used in the approximation. Error bounds for the residuals are established for each case. In the case of power potentials the approximate solutions of an associated terminal value problem at ∞ are shown to be asymptotic power series expansions of the exact solution. Numerical algorithms have been implemented and several examples are given, demonstrating the utility of the approach.