Reconstruction of the potential in the Sturm-Liouville equation with spectral boundary conditions

This study examines the Sturm-Liouville equation, focusing on cases where the spectral parameter is included in the boundary conditions. By employing the Hochstadt-Lieberman theorem and the Weyl function, we demonstrate that when the potential is known for the interval $\left (0,\frac{\pi}{2}\right )$ , a single spectrum can specify the potential for the whole interval $(0,\pi )$ . Furthermore, utilizing the Gesztesy-Simon theorem alongside the Weyl function technique, we establish that if the potential is known a priori on $(0,\pi /2(1 - \alpha ))$ as $\alpha \in (0, 1)$ , some of spectra can uniquely identify the potential across the entire interval $(0, \pi )$ .