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

TL;DR

This paper introduces new theoretical characterizations and a numerical algorithm for estimating spectral density functions of periodic potentials on the half line, with applications to equations like Mathieu's.

Contribution

It provides novel characterizations of spectral functions for Hill's equation and develops a numerical method based on coefficient approximation.

Findings

New characterizations of spectral functions for Hill's equation

Numerical algorithm successfully applied to examples including Mathieu's equation

Demonstrates effectiveness of the method through computational results

Abstract

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.