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Funktsional. Anal. i Prilozhen., 2005, Volume 39, Issue 2, Pages 78–81 (Mi faa44)  

This article is cited in 17 scientific papers (total in 17 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

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


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English version:
Functional Analysis and Its Applications, 2005, 39:2, 148–151

Bibliographic databases:

UDC: 517.5+517.43
Received: 24.07.2003

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

Citation in format AMSBIB
\by V.~N.~Pyvovarchyk
\paper Ambarzumian's Theorem for a Sturm--Liouville Boundary Value Problem on a Star-Shaped Graph
\jour Funktsional. Anal. i Prilozhen.
\yr 2005
\vol 39
\issue 2
\pages 78--81
\jour Funct. Anal. Appl.
\yr 2005
\vol 39
\issue 2
\pages 148--151

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    This publication is cited in the following articles:
    1. Yang Chuan-Fu, Huang Zhen-You, “Inverse spectral problems for $2m$-dimensional canonical Dirac operators”, Inverse Problems, 23:6 (2007), 2565–2574  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Currie S., Watson B.A., “Green's functions and regularized traces of Sturm-Liouville operators on graphs”, Proc. Edinb. Math. Soc. (2), 51:2 (2008), 315–335  crossref  mathscinet  zmath  isi  scopus
    3. Yang Chuan-Fu, Huang Zhen-You, Yang Xiao-Ping, “Ambarzumyan-type theorems for the Sturm-Liouville equation on a graph”, Rocky Mountain J. Math., 39:4 (2009), 1353–1372  crossref  mathscinet  zmath  isi  elib  scopus
    4. Yang Chuan-Fu, Yang Xiao-Ping, “Some Ambarzumyan-type theorems for Dirac operators”, Inverse Problems, 25:9 (2009), 095012, 13 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Brown B.M., Weikard R., “On Inverse Problems for Finite Trees”, Methods of Spectral Analysis in Mathematical Physics, Operator Theory Advances and Applications, 186, 2009, 31–48  mathscinet  zmath  adsnasa  isi
    6. Yang Chuan-Fu, Huang Zhen-You, Yang Xiao-Ping, “Ambarzumyan's theorems for vectorial Sturm-Liouville systems with coupled boundary conditions”, Taiwanese J. Math., 14:4 (2010), 1429–1437  crossref  mathscinet  zmath  isi  scopus
    7. Yang Chuan-Fu, Huang Zhen-You, Yang Xiao-Ping, “Trace formulae for Schrodinger systems on graphs”, Turkish J. Math., 34:2 (2010), 181–196  mathscinet  zmath  isi  elib
    8. Yang Chuan Fu, Pivovarchik V.N., Huang Zhen You, “Ambarzumyan-type theorems on star graphs”, Oper. Matrices, 5:1 (2011), 119–131  crossref  mathscinet  zmath  isi  elib  scopus
    9. 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  crossref  mathscinet  zmath  isi  scopus
    10. Law Ch.-K., Yanagida E., “A Solution to an Ambarzumyan Problem on Trees”, Kodai. Math. J., 35:2 (2012), 358–373  crossref  mathscinet  zmath  isi  scopus
    11. Pivovarchik V., Rozhenko N., “Inverse Sturm-Liouville Problem on Equilateral Regular Tree”, Appl. Anal., 92:4 (2013), 784–798  crossref  mathscinet  zmath  isi  elib  scopus
    12. Davies E.B., “An Inverse Spectral Theorem”, J. Operat. Theor., 69:1 (2013), 195–208  crossref  mathscinet  zmath  isi  scopus
    13. Didenko V.D., Rozhenko N.A., “Inverse Sturm-Liouville Spectral Problem on Symmetric Star-Tree”, Math. Meth. Appl. Sci., 37:15 (2014), 2211–2217  crossref  mathscinet  zmath  isi  scopus
    14. 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  crossref  mathscinet  zmath  isi  elib  scopus
    15. 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  crossref  mathscinet  isi  scopus
    16. Boyko O., Martynyuk O., Pivovarchik V., “Ambarzumian Theorem For Non-Selfadjoint Boundary Value Problems”, J. Operat. Theor., 79:1 (2018), 213–223  crossref  mathscinet  zmath  isi  scopus
    17. Kurasov P., Suhr R., “Schrodinger Operators on Graphs and Geometry. III. General Vertex Conditions and Counterexamples”, J. Math. Phys., 59:10 (2018), 102104  crossref  mathscinet  zmath  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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