Email this article ASYMPTOTICS OF EIGENVALUES AND EIGENFUNCTIONS OF THE STURM–LIOUVILLE OPERATOR WITH SINGULAR POTENTIAL ON A STAR GRAPH. I
- Authors: Zuev K.P1
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Affiliations:
- Lomonosov Moscow State University
- Issue: Vol 61, No 2 (2025)
- Pages: 162-176
- Section: ORDINARY DIFFERENTIAL EQUATIONS
- URL: https://ogarev-online.ru/0374-0641/article/view/299122
- DOI: https://doi.org/10.31857/S0374064125020026
- EDN: https://elibrary.ru/HXKTOT
- ID: 299122
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Abstract
Spectral problems on a star-graph consisting of three edges with a Sturm–Liouville operator defined on each of them are investigated. The spectral properties of such operators have been studied, in particular, asymptotic formulas for eigenvalues and eigenfunctions of the operator with Dirichlet boundary conditions at free ends and continuity and Kirchhoff conditions at a common vertex have been obtained. The potential in the Sturm–Liouville problem is assumed to be singular, it is a derivative of a quadratically summable function in sense of distributions.
References
- Ruedenberg, K. Free-electron network model for conjugated systems. I. Theory / K. Ruedenberg, W.S. Scherr // J. Chem. Physics. — 1953. — V. 21, № 9. — P. 1565–1581.
- Kuchment, P. Graph models for waves in thin structures / P. Kuchment // Waves in Random Media. — 2002. — V. 12, № 4. — P. R1–R24.
- Kuchment, P. Quantum graphs: I. Some basic structures / P. Kuchment // Waves in Random Media. — 2004. — V. 14, № 1. — P. 107–128.
- Proceedings of Symposia in Pure Mathematics. V. 77. Analysis of Graphs and its Applications / Eds. P. Exner, J.P. Keating, P. Kuchment [et al.]. — Cambridge : Amer. Math. Soc., 2007. — 718 p.
- Pokorny, Yu.V., Penkin, O.M., Pryadiev, V.L. [et al.], Differentsial’nyye uravneniya na geometricheskikh grafakh (Differential Equations on Geometrical Graphs), Moscow: Fizmatlit, 2005
- Yurko, V. Inverse spectral problems for Sturm–Liouville operators on graphs / V. Yurko // Inverse Problems. — 2005. — V. 21, № 3. — P. 1075–1086.
- Bondarenko, N. Inverse problems for the differential operator on the graph with a cycle with different orders on different edges / N. Bondarenko // Tamkang J. Math. — 2015. — V. 46, № 3. — P. 229–243.
- Burlutskaya, M.Sh., Zvereva, M.B., and Kamenskii, M.I., Boundary value problem on a geometric star-graph with a nonlinear condition at a node, Math. Notes, 2023, vol. 114, no. 2, pp. 275–279.
- Sadovnichiy, V.A., Sultanaev, Y.T., and Akhtyamov, A.M., Inverse Sturm–Liouville problem with nonseparated boundary conditions on a geometric graph, Differ. Equat., 2019, vol. 55, no. 2, pp. 194–204.
- Zhabko, A.P. Uniqueness solution to the inverse spectral problem with distributed parameters on the graph–star / A.P. Zhabko, K.B. Nurtazina, V.V. Provotorov // Vestnik of Saint Petersburg University. Appl. Math. Comput. Sci. Control Proc. — 2020. — V. 16, № 2. — P. 129–143.
- Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operators with singular potentials, Math. Notes, 1999, vol. 66, no. 6, pp. 741–753.
- Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operators with distribution potentials, Transactions of the Moscow Math. Soc., 2003, vol. 64, pp. 143–192.
- Savchuk, A.M. and Sadovnichaya, I.V., Spectral analysis of one-dimensional Dirac system with summable potential and Sturm–Liouville operator with coefficient–distributions, Sovrem. matematika. Fynd. napravlenia, 2020, vol. 66, no. 3, pp. 373–530.
- Savchuk, A.M., Pryamyye i obratnyye spektral’nyye zadachi dlya operatora Shturma–Liuvillya i sistemy Diraka (Direct and Inverse Spectral Problems for the Sturm–Liouville Operator and the Dirac System), Dissertation for the degree of Doctor of Physical and Mathematical Sciences, 2019.
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