Journal of Geometric Analysis

, Volume 20, Issue 2, pp 439–471 | Cite as

Linear Representations and Isospectrality with Boundary Conditions



We present a method for constructing families of isospectral systems, using linear representations of finite groups. We focus on quantum graphs, for which we give a complete treatment. However, the method presented can be applied to other systems such as manifolds and two-dimensional drums. This is demonstrated by reproducing some known isospectral drums, and new examples are obtained as well. In particular, Sunada’s method (Ann. Math. 121, 169–186, 1985) is a special case of the one presented.


Isospectrality Symmetry Hear the shape of a drum Linear representations Boundary value problem Quantum graphs 

Mathematics Subject Classification (2000)

58J53 58J32 58D19 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. Math. 121, 169–186 (1985) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Kac, M.: Can one hear the shape of a drum? Am. Math. Mon. 73, 1–23 (1966) CrossRefGoogle Scholar
  3. 3.
    Gordon, C., Webb, D., Wolpert, S.: One cannot hear the shape of a drum. Bull. Am. Math. Soc. 27, 134–138 (1992) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gordon, C., Webb, D., Wolpert, S.: Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110, 1–22 (1992) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Buser, P., Conway, J., Doyle, P., Semmler, K.-D.: Some planar isospectral domains. Int. Math. Res. Not. 9, 391–400 (1994) CrossRefMathSciNetGoogle Scholar
  6. 6.
    von Below, J.: Can one hear the shape of a network. In: Partial Differential Equations on Multistructures. Lecture Notes in Pure and Applied Mathematics, vol. 219, pp. 19–36. Dekker, New York (2000) Google Scholar
  7. 7.
    Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Natl. Acad. Sci. USA 51, 542 (1964) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gordon, C., Perry, P., Schueth, D.: Isospectral and isoscattering manifolds: A survey of techniques and examples. Contemp. Math. 387, 157–179 (2005) MathSciNetGoogle Scholar
  9. 9.
    Brooks, R.: Constructing isospectral manifolds. Am. Math. Mon. 95, 823–839 (1988) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brooks, R.: The Sunada method. Contemp. Math. 231, 25–35 (1999) MathSciNetGoogle Scholar
  11. 11.
    Gnutzmann, S., Smilansky, U.: Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006) CrossRefGoogle Scholar
  12. 12.
    Kuchment, P.: Quantum graphs: I. Some basic structures. Waves Random Media 14, S107 (2004) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kuchment, P.: Quantum graphs: An introduction and a brief survey. In: Proc. Symp. Pure Math., pp. 291–314. AMS, Providence (2008) Google Scholar
  14. 14.
    Kostrykin, V., Schrader, R.: Neumann’s rule for quantum wires. J. Phys. A 32, 595–630 (1999) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Band, R., Parzanchevski, O., Ben-Shach, G.: The isospectral fruits of representation theory: Quantum graphs and drums. J. Phys. A 42, 175202 (2009) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gutkin, B., Smilansky, U.: Can one hear the shape of a graph? J. Phys. A 31, 6061–6068 (2001) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Roth, J.P.: Le spectre du Laplacien sur un graphe. In: Proceedings of the Colloque J. Deny Orsay 1983, Lect. Not. Math., vol. 1096, pp. 521–539 (1984) Google Scholar
  18. 18.
    Oren, I.: Private communication (2008) Google Scholar
  19. 19.
    Shapira, T., Smilansky, U.: Quantum graphs which sound the same. In: Khanna, F., Matrasulov, D. (eds.) Nonlinear Dynamics and Fundamental Interactions. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 213, pp. 17–29 (2005) Google Scholar
  20. 20.
    Band, R., Shapira, T., Smilansky, U.: Nodal domains on isospectral quantum graphs: The resolution of isospectrality? J. Phys. A, Math. Gen. 39, 13999–14014 (2006) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Brüning, J., Heintze, E.: Représentations des groupes d’isométries dans les sous-espaces propres du laplacien. C. R. Acad. Sci. Paris 286, 921–923 (1978) MATHGoogle Scholar
  22. 22.
    Bolte, J., Endres, S.: The trace formula for quantum graphs with general self-adjoint boundary conditions. Ann. Henri Poincaré 10(1), 189–223 (2009) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Buser, P.: Isospectral Riemann surfaces. Ann. Inst. Fourier 36, 167–192 (1986) MATHMathSciNetGoogle Scholar
  24. 24.
    Berard, P.: Transplantation et isospectralité I. Math. Ann. 292, 547–559 (1992) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Jakobson, D., Levitin, M., Nadirashvili, N., Polterovich, I.: Spectral problems with mixed Dirichlet-Neumann boundary conditions: Isospectrality and beyond. J. Comput. Appl. Math. 194, 141–155 (2004) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Levitin, M., Parnovski, L., Polterovich, I.: Isospectral domains with mixed boundary conditions. J. Phys. A, Math. Gen. 39, 2073–2082 (2005) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Pesce, H.: Variétés isospectrales et représentations des groupes. Contemp. Math. 173, 231–240 (1994) MathSciNetGoogle Scholar
  28. 28.
    Wigner, E.P.: Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra. Academic Press, New York (1959) Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2009

Authors and Affiliations

  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael
  2. 2.Department of Physics of Complex SystemsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations