Spectral Problems on Compact Graphs

Abstract

The method of nding the eigenvalues and eigenfunctions of abstract discrete semibounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators dened on geometric graphs. The method for nding the eigenfunctions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to nd the eigenvalues and eigenfunctions of the Sturm Liouville operator dened on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high effciency of the developed methods