Communications in Mathematical Physics

, Volume 156, Issue 2, pp 387–397 | Cite as

The integrated density of states for the difference Laplacian on the modified Koch graph

  • Leonid Malozemov


We consider the integrated density of statesN(λ) of the difference Laplacian −Δ on the modified Koch graph. We show thatN(λ) increases only with jumps and a set of jump points ofN(λ) is the set of eigenvalues of −Δ with the infinite multiplicity. We establish also that
$$0< C_1 \leqslant \mathop {\lim }\limits_{\lambda \to 0} \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }}< \overline {\mathop {\lim }\limits_{\lambda \to 0} } \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }} \leqslant C_2< \infty$$
whered s =2log5/log(40/3) is the spectral dimension of MKG.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Spectral Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B] Brolin, H.: Invariant sets under iteration of rational functions. Arkiv for Matematik6, 103–144 (1965)Google Scholar
  2. [F] Fukushima, M.: Dirichlet forms, diffusion processes and spectral dimension for nested fractals. Ideas and Meth. in Math. Anal. Stoch. Appl.1. Cambridge: Cambridge University Press (to appear)Google Scholar
  3. [FS] Fukushima, M., Shima, T.: On a spectral analysis for the Sierpinski gasket. Preprint (1989)Google Scholar
  4. [H] Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J.30, 713–747 (1981)Google Scholar
  5. [K] Kuczma, M.: Functional equations in a single variable. Warszawa: Polish Scientific Publishers 1968Google Scholar
  6. [M] Malozemov, L.A.: Difference Laplacian Δ on the modified Koch curve. Russ. J. Math. Phys.3, 1 (1992)Google Scholar
  7. [R] Rammal, R.: Spectrum of harmonic excitations on fractals. J. Phys.45, 191–206 (1984)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Leonid Malozemov
    • 1
  1. 1.Department of Applied MathematicsMoscow Civil Engineering InstituteMoscowRussia

Personalised recommendations