A solution to the inverse problem for the Sturm-Liouville-type equation with a delay

Abstract

The paper is devoted to study of the inverse problem of the boundary spectral assignment of the Sturm-Liouville with a delay. -y″(x) + q(x)y(α · x) = λy(x); q ∈ AC[0; π];α ∈ (0, 1] (1) with separated boundary conditions: y(0) = y(π) = 0 (2) y(0) = y′(π) = 0 (3) It is argued that if the sequence of eigenvalues is given λn(1) n and λn(2) n tasks (1-2) and (1-3) respectively, then the delay factor α ∈ (0, 1) and the potential q ∈ AC[0, π] are unambiguous. The potential q is composed by means of trigonometric Fourier coefficients. The method can be easily transferred to the case of α = 1 i.e. to the classical Sturm-Liouville problem.