Inverse spectral problems for differential operators on spatial networks

Journal of Inverse and Ill-Posed Problems

2017

Abstract:The Sturm-Liouville operator on a star-shaped graph with different types of boundary conditions (Robin and Dirichlet) in different vertices is studied. Asymptotic formulas for the eigenvalues are derived and partial inverse problems are solved: we show that the potential on one edge can be uniquely determined by different parts of the spectrum if the potentials on the other edges are known. We provide a constructive method for the solution of the inverse problems, based on the Riesz basis property of some systems of vector functions.

Section: Introduction and Main Resultsmentioning

confidence: 99%

Partial inverse problems for the Sturm–Liouville operator on a star-shaped graph with mixed boundary conditions

Journal of Inverse and Ill-Posed Problems

2017

“…One can also develop numerical methods, basing on our algorithm. In addition, we intend to apply our results to inverse problems for differential operators on graphs [39,40].…”

Section: Methodsmentioning

confidence: 99%

Constructive solution of the inverse spectral problem for the matrix Sturm–Liouville operator

Inverse Problems in Science and Engineering

2020

An inverse spectral problem is studied for the matrix Sturm-Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the method of spectral mappings for the considered inverse problem. The nonlinear inverse problem is reduced to a linear equation in a special Banach space of infinite matrix sequences. In addition, we apply our results to the Sturm-Liouville operator on a star-shaped graph.

“…Such operators especially worth to be studied because of their applications to quantum graphs. Inverse problem theory for differential operators on graphs is a rapidly developing field nowadays (see the survey [12]). However, characterization of spectral data is an open problem even for Sturm-Liouville operators on the simplest star-shaped graphs, as well as for the matrix Sturm-Liouville operator with the general self-adjoint boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis of the matrix Sturm–Liouville operator

2019

Bound Value Probl

The self-adjoint matrix Sturm-Liouville operator on a finite interval with a boundary condition in the general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These spectral characteristics play an important role in the inverse spectral theory. Our technique is based on analysis of analytic functions and on the contour integration in the complex plane of the spectral parameter. In addition, we adapt the obtained asymptotic formulas to the Sturm-Liouville operators on a star-shaped graph with two different types of matching conditions.