Kadchenko, Sergeĭ Ivanovich; Stavtseva, Anastasiya Viktorovich; Ryazanova, Lyubov’ Sergeevna; Dubrovskiĭ, Vladislav Vladimirovich Algorithms for the computation of the eigenvalues of discrete semi-bounded operators defined on quantum graphs. (Russian. English summary) Zbl 1511.81056 Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat., Mekh., Fiz. 15, No. 1, 16-25 (2023). Summary: Spectral problems for differential operators defined on quantum graphs are of great scientific interest related to problems in quantum mechanics, computer network modeling, image processing, ranking algorithms, modeling of electrical, and mechanical and acoustic processes, in networks of a diverse nature, in designing nano systems with prescribed properties and in other areas. Theoretical solutions of direct and inverse spectral problems on quantum graphs have been developed, but computational algorithms based on these methods are computationally inefficient. We have not seen any published works that consider examples of numerical solutions of spectral problems on finite connected graphs with a large number of vertices and edges. Therefore, the development of new computationally effective algorithms for numerical solution of spectral problems given on finite connected graphs is urgent. This paper develops a technique for finding the eigenvalues of boundary value problems on finite connected graphs with a required number of vertices and edges. To use this technique, it is necessary to know the eigenvalues and vectors of the eigenfunctions of corresponding unperturbed vector operators which are usually self-adjoint. Finding them manually, if the graph has a large number of vertices and edges, is difficult. This led to writing a package of programs in the mathematical environment Maple to find transcendental equations in the symbolic mode to calculate eigenvalues and find the eigenfunctions of unperturbed boundary value problems. Examples of calculating eigenvalues for a quantum graph which models an anthracene aromatic compound molecule are presented. MSC: 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices Keywords:asymptotic formulas; eigenvalues and eigenfunctions; discrete and self-adjointoperators; inverse spectral problems; Galerkin method Software:Maple × Cite Format Result Cite Review PDF Full Text: DOI MNR References: [1] V.A. Sadovnichii, V.V. Dubrovskii, S.I. Kadchenko, V.F. Kravchenko, “Vychislenie pervykh sobstvennykh chisel kraevoi zadachi gidrodinamicheskoi ustoichivosti mezhdu parallelnymi ploskostyami pri malykh chislakh Reinoldsa”, Doklady Akademii nauk, 355:5 (1997), 605-608 [2] S.I. Kadchenko, “Metod regulyarizovannykh sledov”, Vestnik YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie», 37(170):4 (2009), 4-23 [3] Kadchenko S.I., Kakushkin S.N., “Chislennye metody nakhozhdeniya sobstvennykh chisel i sobstvennykh funktsii vozmuschennykh samosopryazhennykh operatorov”, Vestnik YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie», 27(286):13 (2012), 45-57 [4] S.I. Kadchenko, I.I. Kinzina, “Vychislenie sobstvennykh znachenii vozmuschennykh diskretnykh poluogranichennykh operatorov”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 46:7 (2006), 1265-1272 · Zbl 07811639 [5] S.I. Kadchenko, L.S. Ryazanova, “Chislennyi metod nakhozhdeniya sobstvennykh znachenii diskretnykh poluogranichennykh snizu operatorov”, Vestnik YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie», 17(234):8 (2011), 46-51 · Zbl 1262.47030 [6] S.I. Kadchenko, “Vychislenie ryadov Releya-Shredingera vozmuschennykh samosopryazhennykh operatorov”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 47:9 (2007), 1494-1505 · Zbl 07812377 [7] S.I. Kadchenko, S.N. Kakushkin, “Algoritm nakhozhdeniya sobstvennykh funktsii vozmuschennykh samosopryazhennykh operatorov metodom regulyarizovannykh sledov”, Vestnik YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie», 40(299):14 (2012), 83-88 [8] S.I. Kadchenko, “Chislennyi metod resheniya obratnykh zadach, porozhdennykh vozmuschennymi samosopryazhennymi operatorami, metodom regulyarizovannykh sledov”, Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya, 6(107) (2013), 23-30 [9] V.V. Dubrovskii, S.I. Kadchenko, V.F. Kravchenko, V.A. Sadovnichii, “Computation of the First Eigenvalues of a Discrete Operator”, Elektromagnitnye volny i elektronnye sistemy, 3:2 (1998), 4-7 [10] S.I. Kadchenko, “Chislennyi metod resheniya obratnykh spektralnykh zadach, porozhdennykh vozmuschennymi samosopryazhennymi operatorami”, Vestnik YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie», 6:4 (2013), 15-25 [11] S.I. Kadchenko, S.N. Kakushkin, “Vychislenie znachenii sobstvennykh funktsii diskretnykh poluogranichennykh operatorov metodom regulyarizovannykh sledov”, Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya, 6(97) (2012), 13-21 [12] S.I. Kadchenko, “Algoritm resheniya obratnykh zadach, porozhdennykh vozmuschennymi samosopryazhennymi operatorami”, Aktualnye problemy sovremennoi nauki i tekhniki i obrazovaniya, 3 (2015), 138-141 [13] S.I. Kadchenko, G.A. Zakirova, L.S. Ryazanova, O.A. Torshina, “Obratnaya spektralnaya zadacha opredeleniya neodnorodnosti uprugogo sterzhnya”, Aktualnye problemy sovremennoi nauki i tekhniki i obrazovaniya, 9:2 (2018), 42-45 [14] S.I. Kadchenko, G.A. Zakirova, “A Numerical Method for Inverse Spectral Problems”, Vestnik YuUrGU. Seriya «Matematicheskoe modelirovanie i programmirovanie», 8:3 (2015), 116-126 · Zbl 1344.47015 [15] S.I. Kadchenko, G.A. Zakirova, “Calculation of Eigenvalues of Discrete Semibounded Differential Operators”, Journal of Computational and Engineering Mathematics, 4:1 (2017), 38-47 · Zbl 1541.34105 · doi:10.14529/jcem170104 [16] S.I. Kadchenko, O.A. Torshina, “Vychislenie sobstvennykh chisel ellipticheskikh differentsialnykh operatorov s pomoschyu teorii regulyarizovannykh sledov”, Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Seriya: Matematika. Mekhanika. Fizika, 8 (2016), 36-43 · Zbl 1345.47021 [17] V.V. Dubrovskii, S.I. Kadchenko, V.F. Kravchenko, V.A. Sadovnichii, “Novyi metod priblizhennogo vychisleniya pervykh sobstvennykh chisel spektralnoi zadachi gidrodinamicheskoi teorii ustoichivosti techeniya Puazeilya v krugloi trube”, Doklady Akademii nauk, 380:2 (2001), 160-163 [18] A.V. Stavtseva, Programma resheniya samosopryazhennykh spektralnykh zadach na konechnykh svyazannykh orientirovannykh grafakh, Svidetelstvo № 2021660658 pravoobladatel Stavtseva A.V., zayavlenie 10.06.2021, zaregistrir. 29.07.2021, reestr programmy na EVM This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.