ABSTRACT
In this chapter, we present an alternative approach to the study of multiparameter Sturm-Liouville problems in the case of two equations and two parameters. This is based on a more detailed study of the case of a single boundary-value problem with two parameters. We apply Sturmian methods to the boundaryvalue problem in which we ask whether the equation
y′′(x) + { 2∑
λjpj(x) − q(x)}y(x) = 0, a ≤ x ≤ b, (6.1.1)
has a non-trivial solution satisfying
y(a) cosα = y′(a) sinα, y(b) cosβ = y′(b) sinβ. (6.1.2)
We take p1(x), p2(x) and q(x) to be real and, as a rule, continuous. We take α, β to be real and, without loss of generality, choose them so that
0 ≤ α < π, 0 < β ≤ π. (6.1.3)
Pairs (λ1, λ2) such that (6.1.1)–(6.1.2) have a non-trivial solution may be considered as a sort of eigenvalue. When we are dealing with a single boundaryvalue problem with two parameters, we have no reason to expect that such admissible pairs will have to be real. We shall however confine attention to
spectra.
