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Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 4
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Articles

Inverse Sturm–Liouville problem on equilateral regular tree

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Pages 784-798 | Received 05 Dec 2010, Accepted 08 Nov 2011, Published online: 07 Dec 2011

References

  • Kuchment , P . 2008 . Quantum graphs: an introduction and a brief survey . ‘Analysis on Graphs and its Applications’ Proc. Symp. Pure Math. AMS, N.Y. , : 291 – 314 .
  • Cattaneo , C . 1997 . The spectrum of the continuous Laplacian on a graph . Mh. Math. , 124 : 215 – 235 .
  • Solomyak , M . 2004 . On the spectrum of the Laplacian on regular metric trees . Waves Random Media , 14 : 155 – 171 .
  • von Below , J . 2005 . The eigenvalues of the Laplacian on locally finite networks . Results Math. , 47 : 199 – 225 .
  • Exner , P . 1996 . Weakly coupled states on branching graphs . Lett. Math. Phys. , 38 ( 3 ) : 313 – 320 .
  • Kottos , T and Smilansky , U . 1997 . Quantum chaos on graphs . Phys. Rev. Lett. , 79 : 4794 – 4797 .
  • Carlson , R . 1997 . Hill's equation on a homogeneous tree . Electron. J. Differ. Eqns , 23 : 1 – 30 .
  • Kuchment , P . 2005 . Quantum graphs II: Some spectral properties of quantum and combinatorial graphs . J. Phys. A: Math., Gen. , 38 : 4887 – 4900 .
  • Texier , C and Montambaux , G . 2001 . Scattering theory on graphs . J. Phys. A: Math. Gen. , 34 : 10307 – 10326 .
  • Friedman , J and Tillich , J-P . 2004 . Wave equations for graphs and the edge-based Laplacian . Pac. J. Math. , 216 ( 2 ) : 229 – 266 .
  • Pankrashkin , K . 2006 . Spectra of Schrödinger operators on equilateral quantum graphs . Lett. Math. Phys. , 77 ( 2 ) : 139 – 154 .
  • Pivovarchik , V . 2000 . Inverse problem for the Sturm–Liouville equation on a simple graph . SIAM J. Math. Anal. , 32 : 801 – 819 .
  • Pivovarchik , V . 2007 . Inverse problem for the Sturm–Liouville equation on a star-shaped graph . Math. Nachr. , 13–14 : 1595 – 1619 .
  • Pivovarchik , V . 2005 . Ambarumian's theorem for a Sturm–Liouville boundary value problem on a star-shaped graph . Funct. Anal. Its Appl. , 39 ( 2 ) : 148 – 151 .
  • Yurko , V . 2005 . Inverse spectral problems for Sturm–Liouville operators on graphs . Inverse Probl. , 21 : 1075 – 1086 .
  • Brown , M and Weikard , R . 2005 . A Borg-Levinson theorem for trees . Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. , 461 ( 2062 ) : 3231 – 3243 .
  • Avdonin , S and Kurasov , P . 2008 . Inverse problems for quantum trees . Inverse Probl Imaging , 2 ( 1 ) : 1 – 21 .
  • Currie , S and Watson , B . 2009 . The M-matrix inverse problem for the Sturm–Liouville equation on graphs . Proc. R. Soc. Edinb. Sec A: Math. , 139 : 775 – 796 .
  • Borg , G . 1946 . Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe . Acta Math. , 78 : 1 – 96 .
  • Marchenko , V . 1977 . “ Sturm–Liouville operators and applications (Naukova Dumka) Kiev English trans. ” . In Operator Theory: Advances and Applications, Vol. 22 , Basel : Birkhäuser Verlag . 1986
  • Kostrikin , V and Schrader , R . 1999 . Kirchhoff's rule for quantum wires . J. Phys. A: Math. Gen. , 32 : 595 – 630 .
  • Carlson , R . 1998 . Adjoint and self-adjoint differential operators on graphs . Electron. J. Differ. Eqns , 6 : 10 (electronic)
  • Currie , S and Watson , B . 2005 . Eigenvalue asymptotics for differential operators on graphs . J. Comput. Appl. Math. , 182 : 13 – 31 .
  • Pokornyi , Yu and Pryadiev , V . 2004 . The qualitative Sturm–Liouville theory on spatial networks . J. Mathematical Sciences , 119 ( 6 ) : 788 – 835 .
  • Law , C-K and Pivovarchik , V . 2009 . Characteristic functions of quantum graphs . J. Phys. A: Math. Theor. , 42 : 035302 (11pp)
  • Texier , C . 2008 . On the spectrum of the Laplace operator of metric graphs attached at a vertex-spectral determinant approach . J. Phys. A: Math.Theor. , 41 : 085207
  • Levitan , B . 1987 . Inverse Sturm–Liouville Problems , Moscow, 1984; English translation, VNU Sci. Press, Utrecht : VSP, Zeist Nauka . (in Russian)
  • Levitan , B and Gasymov , M . 1964 . Determination of a differential equation by two of its spectra . Uspekhi Mat. Nauk , 19 ( 2 ) : 3 – 63 . (116) (in Russian)

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