REGULARIZED TRACE OF A SECOND-ORDER DIFFERENTIAL OPERATOR ON A TREE WITH DIRICHLET CONDITIONS

Abstract

This paper considers the problem of calculating the regularized trace of a second-order differential operator on a tree with Dirichlet conditions at the boundary vertices and matching conditions at the interior vertices. The original graph is a tree consisting of seven vertices and six oriented arcs. The relevance of the study is determined by the need to further develop methods of the spectral theory of differential operators on graphs and to generalize known results obtained for operators on a finite interval. To solve the problem, the original tree is divided into two subgraphs, each of which is a star graph. This approach makes it possible to apply established methods of spectral analysis to individual parts of the original graph. Particular attention is paid to constructing the characteristic determinants of the Dirichlet and Neumann problems on the considered subgraphs, as well as to studying their asymptotic behavior. The obtained representations are used to establish asymptotic formulas for the eigenvalues of the original problem. By applying the residue theorem, contour integration methods, and methods of complex analysis, a final formula for the regularized trace of the Sturm–Liouville operator on the considered tree is obtained. The obtained result generalizes known results for differential operators on a finite interval to the case of graph-trees. The study extends existing approaches to the analysis of spectral characteristics of operators on graphs and provides a basis for further investigation of regularized traces on more complex graphs and other types of differential operators.