M-matrix asymptotics for Sturm–Liouville problems on graphs

Abstract

We consider a system formulation for Sturm-Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prüfer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm-Liouville problem on a graph from its M-matrix.

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

“…For formally selfadjoint boundary conditions it was shown in [8, lemma 7.1] that (b) and (c) do not pose additional constraints. In order to define the M -matrix in [10] we needed the two solutions, W 2 and W 3 , of (2.6) such that…”

“…The Titchmarsh-Weyl M -matrix, M = M(λ), of (2.6)-(2.8) was defined in [10] to be the matrix M given by…”

“…In [10] it was shown that the M -matrix defined in (2.12) is a Herglotz function and as such it admits the following representation [16]:…”

“…In § 2, preliminaries from [10] are recalled, while, in § 3, after proving various matrix Wronskian identities and obtaining Green's function for the problem, we show that the poles of the M -matrix are simple, located at the eigenvalues of the boundary-value problem and are on the real axis. It is also shown that the residues at the poles of the M -matrix are negative semi-definite matrices of rank equal to the multiplicity of the eigenvalue (see theorem 3.3).…”

“…This paper is a continuation of [10], in which we considered an M -matrix associated with a system formulation of the Sturm-Liouville operator, with formally self-adjoint boundary conditions, on a graph. There, the M -matrix was related to the matrix Prüfer angle of the system boundary-value problem, and, consequently, with the boundary-value problem on the graph.…”