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.

In this study, we discuss the inverse spectral problem for the energy-dependent Schrödinger equation on a finite interval. We construct an isospectrality problem and obtain some relations between constants in boundary conditions of the problems that constitute isospectrality. Above all, we obtain degeneracy of K(x, t) − K0(x, t) and L(x, t) − L0(x, t) by using a different approach. Some of the main results of our study coincide with results reported by Jodeit and Levitan. However, the method to obtain degeneracy is completely different. Furthermore, we consider all above results for the nonisospectral case.

In this paper, we are concerned with an inverse problem for the Sturm-Liouville operator with Coulomb potential using a new kind of spectral data that is known as nodal points. We give a reconstruction of q as a limit of a sequence of functions whose nth term is dependent only on eigenvalue and its associated nodal data. It is mentioned that this method is based on the works of Law and Yang, but we have applied the method to the singular Sturm-Liouville problem. MSC: 34L05; 45C05

This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the N-fractional calculus operator N ν method, we derive the fractional solutions of the equation.

We study an inverse problem with two given spectra for a second-order differential operator with singularity of the type 2 r + ( + 1) r 2 (here, l is a positive integer or zero) at zero point. It is well known that two spectra {λn} and {μn} uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, π]. One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular, we obtain a new proof of the Hochstadt theorem concerning the structure of the differenceq(r) − q(r).

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Inverse problem for the Bessel operator is studied. A set of values of eigenfunctions at some internal point and parts of two spectra are taken as data. Uniqueness theorems are obtained. The approach that was used in investigation of problems with partially known potential is employed.

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