Abstract:
We study general boundary value problems with nondegenerate characteristic determinant Δ(λ)Δ(λ) for the Sturm–Liouville equation on the interval [0,1][0,1]. Necessary and sufficient conditions for the completeness of root vectors are obtained in terms of the potential. In particular, it is shown that if Δ(λ)≠constΔ(λ)≠const, q(⋅)∈Ck[0,1]q(⋅)∈Ck[0,1] for some k⩾0, and q(k)(0)≠(−1)kq(k)(1), then the system of root vectors is complete and minimal in Lp[0,1] for p∈[1,∞).
Keywords:
Sturm–Liouville equation, completeness of the system of root vectors, nondegenerate boundary conditions.
Citation:
M. M. Malamud, “On the Completeness of the System of Root Vectors of the Sturm–Liouville Operator with General Boundary Conditions”, Funktsional. Anal. i Prilozhen., 42:3 (2008), 45–52; Funct. Anal. Appl., 42:3 (2008), 198–204
\Bibitem{Mal08}
\by M.~M.~Malamud
\paper On the Completeness of the System of Root Vectors of the Sturm--Liouville Operator with General Boundary Conditions
\jour Funktsional. Anal. i Prilozhen.
\yr 2008
\vol 42
\issue 3
\pages 45--52
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\jour Funct. Anal. Appl.
\yr 2008
\vol 42
\issue 3
\pages 198--204
\crossref{https://doi.org/10.1007/s10688-008-0028-0}
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Linking options:
https://www.mathnet.ru/eng/faa2911
https://doi.org/10.4213/faa2911
https://www.mathnet.ru/eng/faa/v42/i3/p45
This publication is cited in the following 29 articles:
Anton A. Lunyov, Mark M. Malamud, “On the completeness property of root vector systems for 2 × 2 Dirac type operators with non-regular boundary conditions”, Journal of Mathematical Analysis and Applications, 543:2 (2025), 128949
A. Lunev, M. Malamud, “On the Asymptotic Expansion of the Characteristic Determinant for a 2 × 2 Dirac Type System”, J Math Sci, 2024
A. A. Lunev, M. M. Malamud, “Ob asimptoticheskom razlozhenii kharakteristicheskogo opredelitelya dlya $2 \times 2$-sistem tipa Diraka”, Issledovaniya po lineinym operatoram i teorii funktsii. 51, Zap. nauchn. sem. POMI, 527, POMI, SPb., 2023, 94–136
Wang Y., “Anh-Version Adaptive Fem For Eigenproblems in System of Second Order Odes: Vector Sturm-Liouville Problems and Free Vibration of Curved Beams”, Eng. Comput., 38:4 (2021), 1807–1830
A. S. Makin, “O dvukhtochechnykh kraevykh zadachakh dlya operatorov Shturma—Liuvillya i Diraka”, Materialy Voronezhskoi vesennei matematicheskoi shkoly
«Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniya–XXX». Voronezh, 3–9 maya 2019 g. Chast 5, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 194, VINITI RAN, M., 2021, 144–154
Akhtyamov A.M., Gasymov T.B., “Degenerated Boundary Conditions of a Sturm-Liouville Problem With a Potential-Distribution”, Azerbaijan J. Math., 11:2 (2021), 98–104
Akhtyamov A.M., “Nonexistence of Degenerate Boundary Conditions in a Spectral Problem”, Differ. Equ., 57:1 (2021), 117–121
Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M., “Degenerate Boundary Conditions For the Diffusion Operator on a Geometric Graph”, Differ. Equ., 56:5 (2020), 595–604
Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M., “Degenerate Boundary Conditions For the Sturm-Liouville Problem on a Geometric Graph”, Differ. Equ., 55:4 (2019), 500–509
Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M., “Degenerate Boundary Conditions on a Geometric Graph”, Dokl. Math., 99:2 (2019), 167–170
Agibalova A.V., Lunyov A.A., Malamud M.M., Oridoroga L.L., “Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated By First Order Systems”, Integr. Equ. Oper. Theory, 91:4 (2019), UNSP 37
A. M. Akhtyamov, “Survey of studies on degenerate boundary conditions and finite spectrum”, Proceedings of the Mavlyutov Institute of Mechanics, 14:3 (2019), 184–291
Makin A., “Two-Point Boundary-Value Problems With Nonclassical Asymptotics on the Spectrum”, Electron. J. Differ. Equ., 2018, 95
A. M. Akhtyamov, “Degenerate boundary conditions for a third-order differential equation”, Differ. Equ., 54:4 (2018), 419–426
Makin A.S., “Basis Properties of the System of Root Functions of the Sturm-Liouville Operator With Degenerate Boundary Conditions: i”, Differ. Equ., 54:10 (2018), 1338–1353
Makin A.S., Moiseev T.E., “Irregular Boundary Value Problem For the Sturm-Liouville Operator”, Differ. Equ., 53:8 (2017), 1021–1028
Makin A.S., “On the absence of the basis property for the root function system of the Sturm–Liouville operator with degenerate boundary conditions”, Dokl. Math., 93:2 (2016), 220–222
Cemile Nur, O. A. Veliev, “On the basis property of the root functions of Sturm–Liouville operators with general regular boundary conditions”, Mosc. Math. J., 15:3 (2015), 511–526
Lunyov A.A., Malamud M.M., “on the Completeness and Riesz Basis Property of Root Subspaces of Boundary Value Problems For First Order Systems and Applications”, J. Spectr. Theory, 5:1 (2015), 17–70
Makin A.S., “On the Completeness of the System of Root Functions of the Sturm-Liouville Operator With Degenerate Boundary Conditions”, Differ. Equ., 50:6 (2014), 835–839
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