Reconstruction of the potential in the Sturm-Liouville equation with spectral boundary conditions - Yasser Khalili; Nematollah Kadkhoda (Journal of Inequalities and Applications)
Abstract 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 ( 0 , π 2 ) $\left (0,\frac{π}{2}\right )$ , a single spectrum can specify the potential for the whole interval ( 0 , π ) $(0,π )$ . Furthermore, utilizing the Gesztesy-Simon theorem alongside the Weyl function technique, we establish that if the potential is known a priori on ( 0 , π / 2 ( 1 − α ) ) $(0,π /2(1 - α ))$ as α ∈ ( 0 , 1 ) $α \in (0, 1)$ , some of spectra can uniquely identify the potential across the entire interval ( 0 , π ) $(0, π )$ .