Numerical methods for solving spectral problems on quantum graphs

S. I. Kadchenko ; A. V. Stavtseva ; L. S. Ryazanova

Journal of computational and engineering mathematics, Tome 8 (2021) no. 3, pp. 49-70.

Résumé

Currently, many authors have developed a number of methods allowing to construct algorithms for the numerical solution of inverse spectral problems. However, from a computational point of view, most of the methods are ineffective, and serious computational difficulties arise in their application. Therefore, the development of new methods for solving spectral problems based on new approaches is urgent. In this article, new algorithms are developed for solving direct and inverse spectral problems defined on quantum graphs. In the developed algorithms, a special role is played by systems of eigenvalues and eigenfunctions of the corresponding unperturbed spectral problem, in which the potentials on all edges of the graph are equal to zero. With a large number of edges in the graph, finding these spectral characteristics faces a large amount of computation. Therefore, in the environment of the Maple package, a registered software package was also written to find the spectral characteristics of unperturbed problems defined on geometric graphs of any configuration and with any finite number of nodes. In the article, the methods for calculating the eigenvalues of discrete problems and solving inverse problems for semi-bounded operators defined on geometric graphs are illustrated by the example of the anthracene molecule. Earlier, on the basis of the numerical methods of regularized traces and the Galerkin method, linear formulas were obtained for calculating the approximate eigenvalues of discrete semi-bounded operators defined on finite intervals. These formulas can be used to find approximate eigenvalues of discrete operators with any ordinal number without using eigenvalues with lower ordinal numbers. This removes many computational difficulties. Using these linear formulas, algorithms are developed for solving direct and inverse problems defined on quantum graphs, which is presented in the article. The constructed algorithm for solving inverse spectral problems defined on sequential geometric graphs with a finite number of links was tested on the anthracene molecule. The algorithm allows to recover the values of unknown functions included in the operators at the discretization nodes using the eigenvalues of the operators and the spectral characteristics of the corresponding self-adjoint operators. The results of numerous experiments show good accuracy and computational efficiency of the developed method.

Détail

Keywords: eigenvalues and eigenfunctions, discrete and self-adjoint operators, inverse spectral problems, Galerkin method, incorrectly set problems, Fredholm integral equation of the first kind, geometric graph.