Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential

Abstract

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

A nodal inverse problem for second order Sturm–Liouville operators with indefinite weights

“…Given q, the determination of a potential q for Àu 00 þ qðxÞu ¼ kqðxÞu; x 2 ½a; b in terms of nodal data has attracted a great deal of interest, see for example [4,9,12,18,14]. Usually, q is assumed constant, although an indefinite problem was studied in [22], with q > 0 in ½a; cÞ and q > 0 in ðc; b.…”

Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

“…Differential equations such as Bessel, hydrogen atom, Hermitte, Jakobi, and Legendre equations can be transformed into Sturm-Liouville equations. There are many studies on these issues [2][3][4][5][6][7]. We also discuss the radial part of Schrödinger's equation for the Bessel equation.…”

“…Fractional calculus is "the theory of derivatives and integrals of any arbitrary real or complex order, which unify and generalize the notions of integer-order differentiation and -fold integration" [6][7][8][9][10][11][12][13]. In recent years, the concept of fractional calculus, originated from Leibniz, has achieved increasing interest during the last two decades.…”

A Hochstadt-Lieberman Theorem for Integro-Differential Operator

“…Whence, there are comperatively little references on inverse problem for integrodifferential operators. Nevertheless, some important results for this operator have been obtained by several authors [4], [16], [17], [18], [19], [20], [21], [22], [23].…”