The isospectrality problem for the classical Sturm-Liouville equation

Abstract

By a classical Sturm-Liouville problem, we mean a two-point boundary value problem of the form \begin{equation} -y''+q(x)y = \lambda y,\quad 0\leq x\leq\pi, \tag{0.1} \end{equation} \begin{equation} y'(0) - hy(0) = 0, \tag{0.2} \end{equation} \begin{equation} y'(\pi)+Hy(\pi) = 0, \tag{0.3} \end{equation} where $q(x)$ is a real continuous function, $h$ and $H$ are real numbers, and $\lambda$ is an eigenvalue. The isospectrality problem is that of describing all problems of the form (0.1)--(0.2)--(0.3) that have the same spectrum. This problem has been studied in detail by Trubowitz et al. in three papers ([1], [2], [3]). Some of the main results of our paper coincide with results in their papers, but the methods are completely different. Our main tools are the Gelfand-Levitan integral equation and transmutation (transformation) operators. The results in the papers [1]--[3] are obtained under the assumption that the potential $q(x)$ belongs to $L^2(0,\pi).$ Instead of this we suppose that the first derivative of $q(x)$ belongs to $L^2(0,\pi).$ This assumption facilitates the proof of the differentiability of the function $\mathcal{F}(x,y)$ (see below) but we omit the proof of differentiability. We hope that this paper will also be useful in studying the cited papers of Trubowitz et al\. and the book [4].