The M-matrix inverse problem for the Sturm—Liouville equation on graphs

Abstract

We consider an inverse spectral problem for Sturm-Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M -matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M -matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M -matrix. This enables us to recover, from the M -matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

A Remark on Inverse Problems for Sturm–Liouville Operators on Graphs

“…The goal of this short note is to compare the M-matrix from [1] with the wellknown spectral data for the classical Sturm-Liouville operators on an interval. Consider the Sturm-Liouville equations − y j (x j ) + q j (x j )y j (x j ) = λy j (x j ), j = 1, s,…”

“…This is the classical inverse Sturm-Liouville problem on an interval. In [1], Eq. (1) is studied on a graph with some boundary conditions.…”

“…(1) is studied on a graph with some boundary conditions. The so-called M-matrix is introduced (see [1] for details), and the uniqueness of recovering equation (1) from the M-matrix is considered. It is easy to check that the specification of the M-matrix from [1] is equivalent to the specification of the Weyl functions m j,ν (λ) for all j = 1, 2s and ν = 0, 1.…”

“…The so-called M-matrix is introduced (see [1] for details), and the uniqueness of recovering equation (1) from the M-matrix is considered. It is easy to check that the specification of the M-matrix from [1] is equivalent to the specification of the Weyl functions m j,ν (λ) for all j = 1, 2s and ν = 0, 1. This means that in [1] there is nothing specific for a graph, and the inverse problem is considered really not on a graph but separately for each fixed j = 1, s on the finite interval (0, l j ).…”

“…It is easy to check that the specification of the M-matrix from [1] is equivalent to the specification of the Weyl functions m j,ν (λ) for all j = 1, 2s and ν = 0, 1. This means that in [1] there is nothing specific for a graph, and the inverse problem is considered really not on a graph but separately for each fixed j = 1, s on the finite interval (0, l j ). In other words, the M-matrix does not reflect spectral properties of operators on a graph.…”