Advances in Differential Equations

The isospectrality problem for the classical Sturm-Liouville equation

Max Jodeit, Jr. and B.M. Levitan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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].

Article information

Adv. Differential Equations, Volume 2, Number 2 (1997), 297-318.

First available in Project Euclid: 24 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A55: Inverse problems 34B05: Linear boundary value problems 34B24: Sturm-Liouville theory [See also 34Lxx] 34L05: General spectral theory 47B06: Riesz operators; eigenvalue distributions; approximation numbers, s- numbers, Kolmogorov numbers, entropy numbers, etc. of operators


Jodeit, Jr., Max; Levitan, B.M. The isospectrality problem for the classical Sturm-Liouville equation. Adv. Differential Equations 2 (1997), no. 2, 297--318.

Export citation