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Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem. [ABSTRACT FROM AUTHOR]

Direct and inverse scattering problem for an operator with non-local potential is solved in the paper. The method is based on the Riemann boundary value problem on a bundle of three straight lines. Description of scattering problem data is given. [ABSTRACT FROM AUTHOR]

Our goal is to find an asymptotic behavior as n → ∞ of the orthogonal polynomials P n (z) defined by Jacobi recurrence coefficients a n (off-diagonal terms) and b n (diagonal terms). We consider the case a n → ∞ , b n → ∞ in such a way that ∑ a n − 1 < ∞ (that is, the Carleman condition is violated) and γ n : = 2 − 1 b n (a n a n − 1) − 1 ∕ 2 → γ as n → ∞. In the case | γ | ≠ 1 asymptotic formulas for P n (z) are known; they depend crucially on the sign of | γ | − 1. We study the critical case | γ | = 1. The formulas obtained are qualitatively different in the cases | γ n | → 1 − 0 and | γ n | → 1 + 0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of P n (z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type. [ABSTRACT FROM AUTHOR]