D.I. Borisov, L.I. Gazizova. Taylor series for resolvents of operators on graphs with small edges ... P. 40-57

We consider a second-order elliptic self-adjoint operator on a graph with small edges. Such a graph is obtained by compressing a given graph by a factor of $\varepsilon^{-1}$ and then gluing it to another fixed graph; here $\varepsilon$ is a small positive parameter. No significant constraints are imposed on this pair of graphs. On such a graph, a general second-order self-adjoint elliptic operator is specified; its differential expression contains derivatives of all orders with variable coefficients and a variable potential. The boundary conditions at the vertices of the graph are also chosen in a general form. All coefficients both in the differential expression and in the boundary conditions can additionally depend on the small parameter $\varepsilon$; this dependence is assumed to be analytical. As was established earlier, the parts of the resolvent of the operator corresponding to the restrictions of the resolvent to the edges of fixed length and to the small edges are analytic in $\varepsilon$ as operators in the corresponding spaces, and the restriction to the small edges should be additionally wrapped by a pair of expansion operators. Analyticity means the possibility to represent these operators in the form of the corresponding Taylor series. The first main result of the paper is a procedure similar to the matching of asymptotic expansions for the recursive determination of all coefficients of these Taylor series. The second main result is the representation of the resolvent by a convergent series similar to a Taylor series with effective estimates of the residuals.

Keywords: graph, small edge, elliptic operator, resolvent, analyticity, Taylor series, matching of asymptotic expansions

Received November 30, 2021

Revised December 17, 2021

Accepted December 27, 2021

Funding Agency: This work was supported by the Russian Science Foundation (project no. 20-11-19995).

Denis Ivanovich Borisov, Dr. Phys.-Math. Sci., Prof., Institute of Mathematics of Ufa Federal Research Center of Russian Academy of Sciences, Ufa, 450008 Russia, e-mail: borisovdi@yandex.ru

Lejsan Ildarovna Gazizova, doctoral student, Institute of Mathematics of Ufa Federal Research Center of Russian Academy of Sciences, Ufa, 450008 Russia, e-mail: gazizovalejsa@gmail.com

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Cite this article as: D.I. Borisov, L.I. Gazizova. Taylor series for resolvents of operators on graphs with small edges, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 40–57; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 317, Suppl. 1, pp. S37–S54.