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 (0,π/2 ) , a single spectrum can specify the potential for the whole interval (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 )) as α ∈ (0, 1) , some of spectra can uniquely identify the potential across the entire interval (0, π ) .