Extremal properties of eigenvalues

In this work, we examine eigenvalues of ordinary and generalized Sturm- Liouville problems. We begin by considering the general boundary value problem -y (t) + q(t)y(t) = λy(t), {cos α)y(a) - {sin α)y\u27(a) = 0, (cos β)y(b) - (sin β)y\u27(b) = 0, where -∞ \u3c a ≤ t ≤ b\u3c ∞, 0 ≤ α \u3c π, 0 \u3c β ≤ π, and q(t) is a real piecewise continuous function. This self-adjoint problem has associated eigenvalues λ0(q) \u3c λ1(q) \u3c ... \u3c λn(q) \u3c ... If we let S := {λo(q) : ∫ab | q(t) | dt = M} where M \u3e 0, our goal is to determine λ0# := sup S and to demonstrate there exists a function q#(t) that satisfies λo(q#) = λ0#. For certain values of α and β, e.g. α = 0 and β = π, this problem has been investigated by several authors. We present a new approach to this problem for general α and β and provide rigorous solutions to cases where only formal solutions have been presented. For 0 ≤ α ≤ π⁄2 and π⁄2 ≤ β ≤ π, we show that λ0# is the root of a transcendental equation and that q#(t) exists. For π⁄2 \u3c α \u3cπ and π⁄2 ≤ β ≤ π, our technique yields a candidate for λ0#, a value λ0** which is a root of a transcendental equation, is an upper bound for S, and is the eigenvalue of the above problem where q contains a delta function. This form of q leads us to consider a more general problem. In the latter half of this work, we consider the generalized differential system dy = (dP) z, dz = (dQ — λdW)y, and establish certain basic results such as the existence and uniqueness of the solution to the initial value problem. For a sequence of initial value problems, we show that the solutions yn and zn converge, the former uniformly and the latter pointwise. Furthermore, the eigenvalues of the boundary value problem are the roots of an entire function; under separated boundary conditions, the eigenvalues are real and simple and a sequence of eigenvalues (corresponding to a sequence of boundary value problems) converge; i.e., the kth eigenvalue of the nth problem converges to the kth eigenvalue of the limit problem as n tends to infinity. This last result enables us to prove that there is a sequence {λ0(n)} in S that converges to λ0** and hence λ0** is the least upper bound of S, i.e., λ0# = λ0**. Moreover, the eigenfunctions (corresponding to the above-mentioned sequence of eigenvalues) also converge, uniformly in yn and pointwise in zn