Advances in Differential Equations

The isospectrality problem for the classical Sturm-Liouville equation

Max Jodeit, Jr. and B.M. Levitan

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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].

Article information

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

Dates
First available in Project Euclid: 24 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366809217

Mathematical Reviews number (MathSciNet)
MR1424771

Zentralblatt MATH identifier
1023.34502

Subjects
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

Citation

Jodeit, Jr., Max; Levitan, B.M. The isospectrality problem for the classical Sturm-Liouville equation. Adv. Differential Equations 2 (1997), no. 2, 297--318. https://projecteuclid.org/euclid.ade/1366809217.


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