Transformation of Sturm–liouville Problems With Decreasing Affine Boundary Conditions

Abstract

We consider Sturm-Liouville boundary-value problems on the interval [0, 1] of the form −y + qy = λy with boundary conditions y (0) sin α = y(0) cos α and y (1) = (aλ + b)y(1), where a < 0. We show that via multiple Crum-Darboux transformations, this boundary-value problem can be transformed 'almost' isospectrally to a boundary-value problem of the same form, but with the boundary condition at x = 1 replaced by y (1) sin β = y(1) cos β, for some β.

Minimality of the system of root functions of Sturm–Liouville problems with decreasing affine boundary conditions

“…In cases (a) and (d), we complete the results of [2] by showing that the system of eigenfunctions of (0.1)-(0.3), with an arbitrary element removed, is minimal in L 2 (0, 1). In cases (b) and (c) we discuss all the choices of the removed element and find necessary and sufficient conditions for the system of root functions, with one element removed, to be minimal in L 2 (0, 1).…”

“…It was proved in [2] (see also [1]) that the eigenvalues of the boundary value problem (0.1)-(0.3) form an infinite sequence accumulating only at ∞ and only the following cases are possible: (a) all eigenvalues are real and simple; (b) all eigenvalues are real and all, except one double, are simple; (c) all eigenvalues are real and all, except one triple, are simple; (d) all eigenvalues are simple and all, except a conjugate pair of non-real ones, are real.…”

“…The asymptotics of eigenvalues and oscillations of eigenfunctions of the boundary value problem (0.1)-(0.3), with the linear function in the boundary condition replaced by a general rational function, were studied in a recent paper [3]. For an affine (linear) decreasing function this asymptotics is as follows [2]:…”

Sturm–Liouville problems with boundary conditions depending quadratically on the eigenparameter

“…The coefficients a, b, c are real and we shall allow a to be either positive or negative. In this sense this paper is both a companion and a sequel to [1] where (3) was replaced by the affine function R(λ) = aλ + b, a < 0. Interest in Sturm-Liouville problems with boundary conditions depending on the eigenparameter has a considerable history which has intensified in recent years with particular attention being paid to cases in which the boundary condition at x = 1 involves various types of functions-see, e.g., [2,4,5,7] where references to earlier work are provided.…”

“…The first contribution to this new area, and the companion to this paper, is to be found in [1], where the thrust is to investigate the use of the Crum-Darboux transformation in the case when R(λ) = aλ + b, a < 0. While the expected complications do arise, the interesting point is that the Crum-Darboux transformation can, with suitable but non-trivial modifications, again be used to produce (almost) isospectral problems with constant boundary conditions thus once more providing new classes of (almost) isospectral problems and also setting the stage for inverse spectral theory.…”