On inverse problem for singular Sturm-Liouville operator from two spectra

Abstract

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).

Explicit Solutions of Fractional SchröDinger Equation via Fractional Calculus Operators

“…provided that the second member of (13) exists. Furthermore, the homogeneous linear ordinary fractional differintegral equation…”

“…For the problem having the analogous singularity, some questions of spectral analysis are given in [13].…”

Scattering theory of Dirac operator with the impulsive condition on whole axis

“…In most applied sciences, especially Sturm–Liouville operators and Dirac operators have been the subject of study for many scientists for many years. It is seen that the spectral and scattering theory of both Sturm–Liouville operators and Dirac operators, with the scalar coefficient, are examined in detail both on the half axis and on the entire axis; see previous works 1‐8 …”

A uniqueness theorem for eigenvalue problem having special potential type

“…Some effective methods of constructing a regular and singular Sturm-Liouville operator for a spectral function or for two spectra are given [2,8,11,19]. We note that the detail of inverse problem for singular equations are given in the monographs and references therein [5,7,9,17,18].…”