A partial inverse Sturm‐Liouville problem on an arbitrary graph

Abstract

The Sturm-Liouville operator with singular potentials of class W −1 2 on a graph of arbitrary geometrical structure is considered. We study the partial inverse problem, which consists in the recovery of the potential on a boundary edge of the graph from a subspectrum under the assumption that the potentials on the other edges are known a priori. We obtain (i) the uniqueness theorem, (ii) a reconstruction algorithm, (iii) global solvability, and (iv) local solvability and stability for this inverse problem. Our method is based on reduction of the partial inverse problem on a graph to the Sturm-Liouville problem on a finite interval with entire analytic functions in the boundary condition. KEYWORDS boundary value problems on graphs and networks for ordinary differential equations, inverse problems involving ordinary differential equations, partial inverse problem, singular potential, Sturm-Liouville operator on graph MSC CLASSIFICATION

On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

“…Arbitrary entire functions in the boundary condition, to the best of our knowledge, for the first time appeared in Yang et al 52 That required, however, making appropriate assumptions on some related functional systems in order to obtain a solution of the inverse problem. Further aspects of such problems were studied in earlier research, 53,54 while their connection with partial inverse problems on geometrical graphs was established in Bondarenko 55 . In our study, this range of questions is addressed for the first time to more general pencil () with nonlinear dependence on the spectral parameter.…”

“…Further aspects of such problems were studied in earlier research, 53,54 while their connection with partial inverse problems on geometrical graphs was established in Bondarenko. 55 In our study, this range of questions is addressed for the first time to more general pencil (1) with nonlinear dependence on the spectral parameter.…”

A new approach to the inverse discrete transmission eigenvalue problem

“…Bondarenko [18][19][20] has developed a unified approach, which is based on the reduction to the form (1.10), to a wide class of inverse problems for differential operators, including the Hochstadt-Lieberman half-inverse problem [21], the inverse transmission eigenvalue problem [1,7], partial inverse problems on metric graphs [20,22]. The approach of [18][19][20] allows not only to prove the uniqueness theorems, but also to develop constructive algorithms for solution and to obtain solvability conditions for various types of partial inverse problems. In this paper, the idea of such reduction is transferred to the discrete case.…”

A new approach to the inverse discrete transmission eigenvalue problem

“…It is worth mentioning that ( 8)-( 9) is the discrete analog of the Sturm-Liouville problem (10) −y + q(x)y = λy, x ∈ (0, π), y(0) = 0, f 1 (λ)y (π) + f 2 (λ)y(π) = 0, with entire analytic functions f 1 (λ) and f 2 (λ) in the boundary condition. Bondarenko [3,4,5] has developed a unified approach, which is based on the reduction to the form (10), to a wide class of inverse problems for differential operators, including the Hochstadt-Lieberman half-inverse problem [16], the inverse transmission eigenvalue problem [19,7], partial inverse problems on metric graphs [6,5]. The approach of [3,4,5] allows not only to prove the uniqueness theorems, but also to develop constructive algorithms for solution and to obtain solvability conditions for various types of partial inverse problems.…”

“…Bondarenko [3,4,5] has developed a unified approach, which is based on the reduction to the form (10), to a wide class of inverse problems for differential operators, including the Hochstadt-Lieberman half-inverse problem [16], the inverse transmission eigenvalue problem [19,7], partial inverse problems on metric graphs [6,5]. The approach of [3,4,5] allows not only to prove the uniqueness theorems, but also to develop constructive algorithms for solution and to obtain solvability conditions for various types of partial inverse problems. In this paper, the idea of such reduction is transferred to the discrete case.…”