Green’s functions and spectral theory for the Hill’s equation

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.

Relationship Between Green’s Functions for Even Order Linear Boundary Value Problems

“…In this section we will show first how the spectra of the considered problems can be connected. The results here generalise those proved in [3]. We will denote by [3], we obtain the following equalities:…”

“…In this section we will use the connecting expressions for Green's functions obtained in Section 3 to compare the values that several Green's functions take point by point. It must be pointed out that, since the relations between the constant sign of the Green's functions are not as strong as for the case n = 1, the results in this section will also be weaker (in some cases) than the ones obtained in [3]. However, some results which could not be deduced for n = 1 hold for n > 1.…”

“…Then, if we denote by G N [T ] and G P [2 T ] the Green's functions related to problems (N, T) and (P, 2 T), respectively, reasoning as in [3], we can deduce that…”

“…The result is the following: Theorem 2. [3,Corollary 4.8] For n = 1 and a 1 ≡ 0, the following properties hold:…”

“…It must be pointed out that the study developed in Sections 3 to 6 has been done in [3] for the particular case of Hill's equation. This is generalized here for any 2n-th order boundary value problem.…”