An inverse spectral problem for the matrix Sturm-Liouville operator on the half-line

Natalia Bondarenko

doi:10.1186/s13661-014-0275-3

Abstract

The matrix Sturm-Liouville operator with an integrable potential on the half-line is considered. The inverse spectral problem is studied, which consists in recovering of this operator by the Weyl matrix. The author provides necessary and sufficient conditions for a meromorphic matrix function being a Weyl matrix of the non-self-adjoint matrix Sturm-Liouville operator. We also investigate the self-adjoint case and obtain the characterization of the spectral data as a corollary of our general result.

Highlights

1 Introduction and main results Inverse spectral problems consist in recovering differential operators from their spectral characteristics
Later Yurko showed that the inverse problem by the generalized spectral function is equivalent to the problem by the generalized Weyl function [ ]
These problems are closely related to the inverse problem for the wave equation utt = uxx – q(x)u

Summary

Introduce the sets

Singularities of the Weyl matrix M(λ) coincide with the zeros of (ρ). Lemma ([ , ]) The Weyl matrix is analytic in outside the countable bounded set of poles , and continuous in outside the bounded set. Let φ(x, λ) = [φjk(x, λ)]mj,k= and S(x, λ) = [Sjk(x, λ)]mj,k= be the matrix solutions of equation ( ) under the initial conditions φ( , λ) = Im, φ ( , λ) = h, S( , λ) = m, S ( , λ) = Im. For each fixed x ≥ , these matrix functions are entire in λ-plane. One can introduce the matrix solutions ∗(x, λ), S∗(x, λ) and φ∗(x, λ) of equation ( ) and the Weyl matrix M∗(λ) := ∗( , λ) of the problem L∗. Let the Weyl matrix M(λ) of the boundary value problem L = L(Q, h) be given.

Introduce auxiliary functions

Im ρ

Taking the relations and

Using the standard estimate