On the determination of the Sturm–Liouville operator from one and two spectra

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 $\widetilde q(x)-q(x)$.
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