Abstract
In this paper, the inverse scattering problem for the Sturm-Liouville operator with discontinuous coefficient and cubic polynomials of the spectral parameter in the boundary condition is considered. The scattering data of the problem is defined, and its properties are investigated. The modified Marchenko main equation is obtained and it is shown that the potential is uniquely recovered by the scattering data.
Highlights
1 Introduction Consider the boundary value problem generated by the differential equation
Inverse spectral problems of spectral analysis often appear in mathematics, mechanics, physics and other branches of natural sciences
The direct scattering problem consists of the determination the collection {S(λ), {λj}Nj=, {mj}Nj= } when q(x) is known
Summary
Where λ is a spectral parameter, q(x) is real valued function with the condition In this paper in the case of discontinuous coefficient ( ), we consider the inverse scattering problem for Sturm-Liouville operator with cubic polynomials of spectral parameter in boundary condition ( ). In the case that ρ(x) ≡ and the boundary condition does not contain a spectral parameter, the inverse scattering problem for ( ) was solved by Marchenko [ , ] and Levitan [ , ].
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