Spectral analysis of the Sturm‐Liouville operator on the star‐shaped graph

Abstract

The inverse spectral problems are studied for the Sturm-Liouville operator on the star-shaped graph and for the matrix Sturm-Liouville operator with the boundary condition in the general self-adjoint form. We obtain necessary and sufficient conditions of solvability for these two inverse problems, and also prove their local solvability and stability.

Weight matrices of the Dirac operator on a star graph

“…Bondarenko 12 further obtained the necessary and sufficient conditions for the solvability of this inverse problem. Recently, Bondarenko 13,14 also extended above results to Sturm-Liouville operators on star graphs.…”

“…In the present paper, based on the method of work, 13 we plan to define the weight matrices of the Dirac operator on a star graph and investigate their structure properties and asymptotic formulae. Furthermore, we use these properties to find a Riesz basis consisting of eigenfunctions of the operator.…”

Inverse problem solution and spectral data characterization for the matrix Sturm-Liouville operator with singular potential

“…The issues of constructive solution and spectral data characterization for this operator appeared to be more difficult for investigation because of complex asymptotic behavior of the spectrum and structural properties of the problem. In [27,28], properties of the spectral data have been investigated for the matrix Sturm-Liouville operator with boundary condition in the general selfadjoint form at x = π and with Dirichlet boundary condition at x = 0. Further, a constructive solution procedure has been developed for the corresponding inverse spectral problem (see [29]).…”