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On the determination of the Sturm–Liouville operator from one and two spectra
B. M. Levitan
Abstract:
Let the sequences
$\{\lambda_n\}_0^\infty$ and
$\{\mu_n\}_0^\infty$ define the Sturm–Liouville problem
\begin{equation}
\tag{I}
\left.\begin{gathered}
-y''+\{\lambda-q(x)\}y=0\quad(0\leqslant x\leqslant\pi),\\
y'(0)-hy(0)=0,\quad y'(\pi)+Hy(\pi)=0,
\end{gathered}\right\}
\end{equation}
and, in addition, let the sequences $\{\widetilde\lambda_n\}_0^\infty=\{\lambda_n\}_0^\infty$ and
$\{\widetilde\mu_n\}_0^\infty$, where
$\widetilde\mu_n=\mu_n$ for
$n>N\geqslant0$,
define a second Sturm–Liouville problem
\begin{equation}
\tag{II}
\left.\begin{gathered}
-y''+\{\lambda-\widetilde q(x)\}y=0,\\
y'(0)-\widetilde hy(0)=0,\quad y'(\pi)+\widetilde Hy(\pi)=0.
\end{gathered}\right\}
\end{equation}
In this paper we show that the kernel
$F(x,s)$ of the integral equation for the inverse problem, in which problem (II) is regarded as a perturbation of problem (I), has the form
$$
F(x,s)=\sum_{n=0}^N\psi(x,\widetilde\mu_n)\varphi(s,\widetilde\mu_n),
$$
in the triangle
$0\leqslant s\leqslant x\leqslant\pi$, wherein
$\psi(x,\lambda)$ and
$\varphi(s,\lambda)$ are solutions of (I). In particular, we obtain a new proof of Hochstadt's theorem concerning the structure of the difference
$\widetilde q(x)-q(x)$.
Bibliography: 5 titles.
UDC:
517.9
MSC: Primary
34B25; Secondary
45A05 Received: 13.09.1976