Half-inverse problem for diffusion operators on the finite interval

Abstract

The potential function q ( x ) in the regular and singular Sturm–Liouville problem can be uniquely determined from two spectra. Inverse problem for diffusion operator given at the finite interval eigenvalues, normal numbers also on two spectra are solved. Half-inverse spectral problem for a Sturm–Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. In this study, by using the Hochstadt and Lieberman's method we show that if q ( x ) is prescribed on [ π 2 , π ] , then only one spectrum is sufficient to determine q ( x ) on the interval [ 0 , π 2 ] for diffusion operator.