Algorithms for estimating spectral density functions for periodic potentials on the half line

For Hill's equation on [0,infinity) we prove new characterizations of the spectral function rho(lambda) and the spectral density function f(lambda) based on analysis involving a companion system of first order differential equations in [6,7]. A numerical algorithm is derived and implemented based on coefficient approximation. Results for several examples, including the Mathieu equation, are presented.