Eigencurves for Two-Parameter Sturm-Liouville Equations

Abstract

This paper concerns two-parameter Sturm-Liouville problems of the form -(p(x)yl) + q(x)y (kr(x) + lz)y, a < x < b with self-adjoint boundary conditions at a and b. The set of (),/) 6 R for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions. Our focus is on these graphs, which are termed eigencurves in the literature.Although eigencurves have been used in a variety of ways for about a century, they seem comparatively underdeveloped in their own fight. Our plan is to give motivation for the topic, elementary properties of eigencurves, illustrations on a simple example first studied by Richardson in 1918 (and since then by several authors), and sone natural questions which may whet the reader's appetite. Some of these questions lead to new types of inverse Sturm-Liouville problems.

Non-real eigenvalues of singular indefinite Sturm-Liouville operators

“…the real spectrum of A necessarily accumulates to +∞ and −∞ and A may have non-real eigenvalues which possibly accumulate to the real axis (see [3,4,9,14,16,20]). For further indefinite Sturm-Liouville problems, applications and references, see, e.g., [2,6,7,11,13,15,23,26].…”

Spectral inversion of an indefinite Sturm–Liouville problem due to Richardson

“…The minima do not form such a lattice and their positions and values have been determined numerically. Binding and Volkmer [3] have argued and to some extent proved that there are complex connections between the eigencurves of figure 1. We wish to elaborate on this and argue that all of the eigencurves are different branches of the same function and that the complex connections are facilitated by square root branch points off the real axis of λ.…”

“…In the mathematical literature there is some discussion of these questions at an abstract [7] as well as at a more concrete [3] level.…”

“…The two integers indicate the sheets that are joined at the square root. E n (λ) is subject to two constraints: since E n (λ) is real on the real axis we have E n (λ * ) = E * n (λ), and in [3] it is proved that E n (−λ) = E n (λ). These conditions require that branch points on the imaginary axis appear as mirror pairs and also that those in the first quadrant at λ = x + iy must appear at the four positions λ = ±x ± iy.…”

“…The history of this early period has been reviewed by Mingarelli [2] with later developments summarized by Binding and Volkmer [3]. The spectrum of the Richardson problem depends on a real parameter in a very complicated way that has never been fully understood and has even been called amazing [4].…”