Green's Functions and Spectral Theory for the Hill's Equation

TL;DR

This paper explores properties of Green's functions for Hill's equation under various boundary conditions, deriving explicit formulas and spectral results to compare solutions across different boundary value problems.

Contribution

It provides explicit expressions for Green's functions of Hill's equation with various boundary conditions and establishes spectral and comparison results.

Findings

Explicit formulas for Green's functions under different boundary conditions

Spectral properties related to Hill's equation

Comparison results for solutions with different boundary conditions

Abstract

The aim of this paper is to show certain properties of the Green's functions related to the Hill's equation coupled with different two point boundary value conditions. We will obtain the expression of the Green's function of Neumann, Dirichlet, Mixed and anti-periodic problems as a combination of the Green's function related to periodic ones. As a consequence we will prove suitable results in spectral theory and deduce some comparison results for the solutions of the Hill's equation with different boundary value conditions.