Generalizations of Borg’s uniqueness theorem to the case of nonseparated boundary conditions

Abstract

Inverse Sturm–Liouville problems and generalizations of Borg’s uniqueness theorem to the case of general boundary conditions are considered. New generalizations of Borg, Marchenko and Karaseva’s uniqueness theorems to the case of nonseparated boundary conditions are obtained. Appropriate examples and a counterexample are given. For illustration we recall Ambarzumijan’s setting \( -y'' + q(x)\,y = \lambda y \) with boundary conditions \( y'(0) = y'(\pi) = 0 \), where \( \int_{0}^{\pi} q(x)\,dx = 0 \) and the eigenvalues are \( 1^2, 2^2, \ldots \). We also mention Borg’s problem with boundary conditions \( y'(0) - h\,y(0) = 0 \) and \( y'(\pi) + H\,y(\pi) = 0 \).