## Advances in Differential Equations

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

### The isospectrality problem for the classical Sturm-Liouville equation

Max Jodeit, Jr. and B.M. Levitan

#### 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