Spectral properties of Schrödinger operators on radial N-dimensional infinite trees

  • Yehuda PinchoverEmail author
  • Gershon Wolansky
  • Daphne Zelig


We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.


Weight Function Laplace Operator Discrete Spectrum Lipschitz Domain Counting Function 
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  1. [1]
    D. E. Anagnostou, M. T. Chryssomallis, J. C. Lyke and C. G. Christodoulou, Improved multiband performance with self-similar fractal antennas, in IEEE ASP Tropical Conference on Wireless Communications Technology, Honolulu, 2003, pp. 271–272.Google Scholar
  2. [2]
    H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman Advanced Publishing Program, Boston, 1984.Google Scholar
  3. [3]
    M. Sh. Birman and V. V. Borzov, The asymptotic behavior of the discrete spectrum of certain singular differential operators, in Problems of Mathematical Physics, No. 5: Spectral Theory (Russian), Izdat. Leningrad. Univ., Leningrad, 1971, pp. 24–38; English transl. in: Spectral Theory, M. Sh. Birman (Ed.), Topics in Mathematical Physics, Vol. 5., Consultants Bureau, New York, 1972.Google Scholar
  4. [4]
    R. Carlson, Nonclassical Sturm-Liouville problems and Schrödinger operators on radial trees, Electronical Journal of Differential Equations 2000 (2000), 1–24.Google Scholar
  5. [5]
    W. D. Evans and Y. Saito, Neumann Laplacians on domains and operators on associated trees, The Quarterly Journal of Mathematics 51 (2000), 313–342.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    T. Kato, Pertubation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.Google Scholar
  7. [7]
    S. Kosugi, A semilinear elliptic equation in a thin network-shaped domain, Journal of the Mathematical Society of Japan 52 (2000), 673–697.zbMATHMathSciNetGoogle Scholar
  8. [8]
    S. Kosugi, Semilinear elliptic equations on thin network-shaped domain with variable thickness, Journal of Differential Equations 183 (2002), 165–188.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    P. Kuchment and H. Zeng, Convergence of spectra of mesoscopic systems collapsing onto a graph, Journal of Mathematical Analysis and Applications 258 (2001), 671–700.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. Kuchment and H. Zeng, Asymptotics of spectra of Neumann Laplacians in thin domains, in Advances in Differential Equations and Mathematical Physics, (Yu. Karpeshina, G. Stolz, R. Weikard and Y. Zeng, eds.), Contemporary Mathematics AMS 387, 2003, pp. 199–213.Google Scholar
  11. [11]
    R. T. Lewis, Singular elliptic operators of second order with purely discrete spectra, Transactions of the American Mathematical Society 271 (1982), 653–666.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., New York, 1982.zbMATHGoogle Scholar
  13. [13]
    V. G. Maz’ja, Sobolev Spaces, Springer-Verlag, Berlin, 1985.zbMATHGoogle Scholar
  14. [14]
    K. Naimark, and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proceedings of the London Mathematical Society 3 (2000), 690–724.CrossRefMathSciNetGoogle Scholar
  15. [15]
    K. Naimark, and M. Solomyak, Geometry of Sobolev spaces on regular trees and the Hardy inequalities, Russian Journal of Mathemaatical Physics 8 (2001), 322–335.zbMATHMathSciNetGoogle Scholar
  16. [16]
    T. R. Nelson, and D. K. Manchester, Modeling of lung morphogenesis using fractal geometries, IEEE Transactions of Medical Imaging 7 (1988), 321–327.CrossRefGoogle Scholar
  17. [17]
    T. R. Nelson, B. J. West, and A. L. Goldberger, The fractal lung: Universal and species-related scaling patterns, Experientia 46 (1990), 251–254.CrossRefGoogle Scholar
  18. [18]
    O. Post, Spectral convergence of non-compact quasi-one-dimensional spaces, Annales Henri Poincaré 7 (2006), 933–973.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    C. Puente, J. Claret, F. Sagués, J. Romeu, M. Q. López-Salvans and R. Pous, Multiband properties of a fractal tree antenna generated by electrochemical deposition, IEE Electronics Letters 32, (1996), 2298–2299.CrossRefGoogle Scholar
  20. [20]
    C. Puente-Baliarda, J. Romeu, R. Pous and A. Cardama, On the behavior of the Sierpinski multiband fractal antenna, IEEE Transactions on Antennas Propagation 46 (1998), 517–524.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV, Academic Press, New York, 1980.zbMATHGoogle Scholar
  22. [22]
    J. Rubinstein and M. Schatzman, Variational problems on multiply connected thin strip I: Basic estimates and convergence of the Laplacian spectrum, Archive for Rational Mechanics and Analysis 160 (2001), 271–308.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    M. Solomyak, Laplace and Schrödinger operators on regular metric trees: the discrete spectrum case, in Function Spaces, Differential Operators and Nonlinear Analysis, The Hans Triebel Anniversary Volume, Birkhäuser, Basel, 2003, pp. 161–181.Google Scholar
  24. [24]
    M. Solomyak, On the eigenvalue estimates for the weighted Laplacian on metric graphs, in Nonlinear Problems in Mathematical Physics and Related Topics I, In Honor of Professor O. A. Ladyzhenskaya, (M. Sh. Birman, S. Hildebrandt, V. A. Solonnikov and N. H. Uraltseva, eds.), Kluwer, New York, 2002, pp. 327–347.Google Scholar
  25. [25]
    E. R. Weibel, Design of airways and blood vessels considered as branching tree, Chapter 74, in The Lung, (R. G. Crystal, J. B. West and P. J. Barnes, eds.), Lippencott-Raven Inc., Philadelphia, 1997.Google Scholar
  26. [26]
    D. Zelig, Properties of solutions of partial differential equations defined on human lung-shaped domains, Ph. D thesis, Technion-Israel Institute of Technology, Israel, 2005.Google Scholar

Copyright information

© The Hebrew University of Jerusalem 2008

Authors and Affiliations

  • Yehuda Pinchover
    • 1
    Email author
  • Gershon Wolansky
    • 1
  • Daphne Zelig
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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