The inverse problem of the scattering theory for Sturm-Liouville operator on the half line with boundary condition depending quadratic on the spectral parameter is considered. Scattering data are defined, some properties of the scattering data are examined, the main equation is obtained, solvability of the integral equation is proved and uniqueness of algorithm to the potential with given scattering data is studied.

In the present study, we investigate the existence of spectral functions and obtain the Parseval identity and expansion formula in eigenfunctions for the singular q-Sturm–Liouville problem with transmission conditions.

This study is based upon investigations of the existence of a singular Hahn difference equation of q,ω$$ q,\omega $$‐Sturm–Liouville problem with transmission conditions. Moreover, the Parseval identity and expansion formula in eigenfunctions are obtained.

Communicated by A. KirschWe considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein-Gordon differential equation with a nonlinear spectral parameter-dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S. /; in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson-type formula, was obtained. Copyright