V. N. Pyvovarchyk, “Ambarzumian's Theorem for a Sturm–Liouville Boundary Value Problem on a Star-Shaped Graph”, Funktsional. Anal. i Prilozhen., 39:2 (2005), 78–81; Funct. Anal. Appl., 39:2 (2005), 148

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Funktsional'nyi Analiz i ego Prilozheniya, 2005, Volume 39, Issue 2, Pages 78–81
DOI: https://doi.org/10.4213/faa44 (Mi faa44)

This article is cited in 28 scientific papers (total in 28 papers)

Brief communications

Ambarzumian's Theorem for a Sturm–Liouville Boundary Value Problem on a Star-Shaped Graph

V. N. Pyvovarchyk

Odessa State Academy of Building and Architecture

Full-text PDF (167 kB) Citations (28)

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DOI: https://doi.org/10.4213/faa44

Abstract: Ambarzumian's theorem describes the exceptional case in which the spectrum of a single Sturm–Liouville problem on a finite interval uniquely determines the potential. In this paper, an analog of Ambarzumian's theorem is proved for the case of a Sturm–Liouville problem on a compact star-shaped graph. This case is also exceptional and corresponds to the Neumann boundary conditions at the pendant vertices and zero potentials on the edges.

Keywords: inverse problem, Neumann boundary conditions, normal eigenvalue, multiplicity of an eigenvalue, least eigenvalue, minimax principle.

Received: 24.07.2003

English version:
Functional Analysis and Its Applications, 2005, Volume 39, Issue 2, Pages 148–151
DOI: https://doi.org/10.1007/s10688-005-0029-1

Bibliographic databases:

Document Type: Article

UDC: 517.5+517.43

Language: Russian

Citation: V. N. Pyvovarchyk, “Ambarzumian's Theorem for a Sturm–Liouville Boundary Value Problem on a Star-Shaped Graph”, Funktsional. Anal. i Prilozhen., 39:2 (2005), 78–81; Funct. Anal. Appl., 39:2 (2005), 148–151

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https://doi.org/10.4213/faa44

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This publication is cited in the following 28 articles:

Dong-Jie Wu, Xin-Jian Xu, Chuan-Fu Yang, “Ambarzumyan's Theorem for the Dirac Operator on Equilateral Tree Graphs”, Acta Math. Appl. Sin. Engl. Ser., 40:2 (2024), 568
Patrizio Bifulco, Joachim Kerner, “A note on Ambarzumian's theorem for quantum graphs”, Arch. Math., 123:1 (2024), 95
M. Kiss, “An Ambarzumian type theorem on graphs with odd cycles”, Ukr. Mat. Zhurn., 74:12 (2023), 1679
M. Kiss, “An Ambarzumian-Type Theorem on Graphs with Odd Cycles”, Ukr Math J, 74:12 (2023), 1916
Zhang R., Yang Ch.-F., “Ambarzumyan-Type Theorem For the Impulsive Sturm-Liouville Operator”, J. Inverse Ill-Posed Probl., 29:1 (2021), 21–25
Yang Ch.-f., Wang F., Huang Zh.-y., “Ambarzumyan Theorems For Dirac Operators”, Acta Math. Appl. Sin.-Engl. Ser., 37:2 (2021), 287–298
Kurasov P., Suhr R., “Asymptotically Isospectral Quantum Graphs and Generalised Trigonometric Polynomials”, J. Math. Anal. Appl., 488:1 (2020), 124049
Chernyshenko A., Pivovarchik V., “Recovering the Shape of a Quantum Graph”, Integr. Equ. Oper. Theory, 92:3 (2020), 23
Kiss M., “Spectral Determinants and An Ambarzumian Type Theorem on Graphs”, Integr. Equ. Oper. Theory, 92:3 (2020), 24
Zhang W., Liu W., “Existence and Ulam'S Type Stability Results For a Class of Fractional Boundary Value Problems on a Star Graph”, Math. Meth. Appl. Sci., 43:15 (2020), 8568–8594
Boman J., Kurasov P., Suhr R., “Schrodinger Operators on Graphs and Geometry II. Spectral Estimates For l-1-Potentials and An Ambartsumian Theorem”, Integr. Equ. Oper. Theory, 90:3 (2018), UNSP 40
Boyko O., Martynyuk O., Pivovarchik V., “Ambarzumian Theorem For Non-Selfadjoint Boundary Value Problems”, J. Operat. Theor., 79:1 (2018), 213–223
Kurasov P., Suhr R., “Schrodinger Operators on Graphs and Geometry. III. General Vertex Conditions and Counterexamples”, J. Math. Phys., 59:10 (2018), 102104
Yang Ch.-F., Xu X.-Ch., “Ambarzumyan-type theorems on graphs with loops and double edges”, J. Math. Anal. Appl., 444:2 (2016), 1348–1358
Didenko V.D., Rozhenko N.A., “Inverse Sturm-Liouville Spectral Problem on Symmetric Star-Tree”, Math. Meth. Appl. Sci., 37:15 (2014), 2211–2217
Pivovarchik V., Rozhenko N., “Inverse Sturm-Liouville Problem on Equilateral Regular Tree”, Appl. Anal., 92:4 (2013), 784–798
Davies E.B., “An Inverse Spectral Theorem”, J. Operat. Theor., 69:1 (2013), 195–208
Law Ch.-K., Yanagida E., “A Solution to an Ambarzumyan Problem on Trees”, Kodai. Math. J., 35:2 (2012), 358–373
Yang Chuan Fu, Pivovarchik V.N., Huang Zhen You, “Ambarzumyan-type theorems on star graphs”, Oper. Matrices, 5:1 (2011), 119–131
Yang Chuanfu, Yang Xiaoping, “Ambarzumyan's Theorem With Eigenparameter in the Boundary Conditions”, Acta Math Sci Ser B Engl Ed, 31:4 (2011), 1561–1568

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Functional Analysis and Its Applications

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