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'(a) = 0, (cos β)y(b) - (sin β)y'(b) = 0, where -∞ < a ≤ t ≤ b< ∞, 0 ≤ α < π, 0 < β ≤ π, and q(t) is a real piecewise continuous function. This self-adjoint problem has associated eigenvalues λ₀(q) < λ₁(q) < ... < λ_n(q) < ... If we let S := {λ_o(q) : ∫_a^b | q(t) | dt = M} where M > 0, our goal is to determine λ₀^# := sup S and to demonstrate there exists a function q^#(t) that satisfies λ_o(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 < α <π 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 y_n and z_n 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 y_n and pointwise in z_n.