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Articles

Location and multiplicities of eigenvalues for a star graph of Stieltjes strings

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Pages 383-402 | Received 13 Aug 2014, Accepted 07 Nov 2014, Published online: 23 Mar 2015

References

  • O.Boyko, and V.Pivovarchik, Inverse spectral problem for a star graph of Stieltjes strings, Methods Funct. Anal. Topology14(2) (2008), pp. 159–167.
  • B.M.Brown, M.S.P.Eastham, and I.G.Wood, Estimates for the lowest eigenvalue of a star graph, J. Math. Anal. Appl.354(1) (2009), pp. 24–30.
  • S.J.Cox, M.Embree, and J.M.Hokanson, One can hear the composition of a string: experiments with an inverse eigenvalue problem, SIAM Rev.54(1) (2012), pp. 157–178.
  • S.J.Cox, M.Embree, and J.M.Hokanson, Inverse eigenvalue experiments for beaded strings. 2010. http://www.caam.rice.edu/∼beads.
  • W.Cauer, Die Verwirklichung von Wechselstromwiderständen vorgeschriebener Frequenzabhängigkeit, Archiv f. Elektrotechnik17(4) (1926), pp. 355–388.
  • P.Exner, Bound states and resonances in quantum wires, Oper. Theory Adv. Appl.46 (1990), pp. 65–84.
  • A.F.Filimonov, and A.D.Myshkis, On properties of large wave effect in classical problem of bead string vibration, J. Difference Equ. Appl.10(13–15) (2004), pp. 1171–1175.
  • F.R.Gantmakher, and M.G.Krein, Oscillating Matrices and Kernels and Vibrations of Mechanical Systems (in Russian) in GITTL, Moscow-Leningrad, (1950), German transl, Akademie Verlag, Berlin, 1960.
  • G.Gladwell, Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, 2004.
  • G.Gladwell, Matrix inverse eigenvalue problems. In: G. Gladwell, A. Morassi, eds., Dynamical Inverse Problems: Theory and ApplicationsCISM Courses and Lectures529 (2011), pp. 1–29.
  • B.J.Gómez, C.E.Repetto, C.R.Stia, and R.Welti, Oscillations of a string with concentrated masses, Eur. J. Phys.28 (2007), pp. 961–975.
  • G.H.Hardy, J.E.Littlewood, and G.Pólya, Inequalities, Second Edition, Cambridge University Press, Cambridge, 1952, p. xii+324.
  • C.R.Johnson, and A.Leal Duarte, On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree, Linear Algebra Appl348 (2002), pp. 7–21.
  • I.S.Kac, and M.G.Krein, On the spectral function of the string, Amer. Math. Soc. Transl.103(2) (1974), pp. 19–102.
  • M.G.Krein, and H.Langer, Continuation of Hermitian positive definite functions and related questions, Integral Equations Operator Theory78(1) (2014), pp. 1–69.
  • P.Kurasov, Can one distinguish quantum trees from the boundary?Proc. Amer. Math. Soc.140(7) (2012), pp. 2347–2356.
  • C.-K.Law, V.Pivovarchik, and W.C.Wang, A polynomial identity and its applications to inverse spectral problems in Stieltjes strings, Operators Matrices7(3) (2013), pp. 603–617.
  • V.A.Marchenko, Introduction to the Theory of Inverse Problems of Spectral Analysis (in Russian)Acta, Kharkov, 2005.
  • A.Marshall, I.Olkin, and B.Arnold, Inequalities: Theory of Majorization and Its Applications, Second Edition, Springer, New York, 2011, p. xxviii+909.
  • R.F.Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc. Edinburgh Math. Soc.21 (1903), pp. 144–157.
  • V.Pivovarchik, N.Rozhenko, and C.Tretter, Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings, Linear Algebra Appl.439(8) (2013), pp. 2263–2292 (open access).
  • V.Pivovarchik, and H.Woracek, Sums of Nevanlinna functions and differential equations on star-shaped graphs, Operators Matrices3(4) (2009), pp. 451–501.
  • T.-L.Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, 8 (1894), 1–122, 9 (1895), 1–47.
  • G.Valent, and W.Van Assche, The impact of Stieltjes' work on continued fractions and orthogonal polynomials: additional material, J. Comput. Appl. Math65(1–3) (1995), pp. 419–447.
  • W.Van Assche, The impact of Stieltjes work on continued fractions and orthogonal polynomials, in Œuvres complètes/Collected papers, Thomas JanStieltjes, ed., Vol. I, II. Reprint of the 1914-1918 editionSpringer-Verlag, Berlin, 1993, pp. 5–37.