On the inverse problem for Sturm–Liouville-type operators with frozen argument: rational case

Abstract

We study two inverse problems of recovering the complex-valued squareintegrable potential q(x) in the operator −y ′′ (x) + q(x)y(a), y (ν) (0) = γy (ν) (1) for ν = 0, 1, where γ ∈ C \ {0}. The first problem involves only single spectrum as the input data. We obtain complete characterization of the spectrum and prove that its specification determines q(x) uniquely if and only if γ = ±1. For the rest cases: periodic and antiperiodic ones, we describe classes of iso-spectral potentials and provide restrictions under which the uniqueness holds. The second inverse problem deals with recovering q(x) from the two spectra related to γ = ±1. We obtain necessary and sufficient conditions for its solvability and establish that uniqueness holds if and only if a = 0, 1. For a ∈ (0, 1), we describe classes of iso-bispectral potentials and provide restrictions under which the uniqueness resumes. Algorithms for solving both inverse problems are provided.

Finite-difference approximation of the inverse Sturm-Liouville problem with frozen argument

“…for various types of boundary conditions (the Neumann BC, periodic BC, etc. ; see [7,23,24,26]). Those types of boundary conditions imply a richer structure of degenerate and non-degenerate cases than the Dirichlet ones.…”

“…Inverse problems for the functional-differential equation (1.3) were studied in [7,[21][22][23][24][25][26][27][28][29]. In particular, it has been shown in [23] that the spectrum of the problem (1.3)- (1.4) is the countable set of the eigenvalues {λ n } ∞ n=1 (counting with multiplicities) with the asymptotics…”

“…Degenerate case (see [23,24,26,29]). If π a ∈ Q, that is, π a = j k , j, k ∈ N, j < k, then a part of the spectrum degenerates:…”

“…In this case, the potential q(x) cannot be uniquely recovered from the spectrum {λ n } ∞ n=1 . However, in [26], the classes of iso-spectral potentials are described and additional restrictions sufficient for the uniqueness of the inverse problem solution are provided. Furthermore, in [23,24,26] constructive algorithms for solving the inverse problems together with the necessary and sufficient conditions of their solvability are obtained.…”

An inverse problem for an integro‐differential operator on a star‐shaped graph

“…VI. Using the methods of this paper, one can obtain the results of the studies [34][35][36]39] concerning the inverse problems for the functional-differential operators with frozen argument on a finite interval in another way. Let us describe the main idea.…”