Volume 88, Number 4, November 2009
Article Number 40007
Number of page(s) 5
Section General
Published online 03 December 2009
EPL, 88 (2009) 40007
DOI: 10.1209/0295-5075/88/40007

Physical consequences of complex dimensions of fractals

E. Akkermans1, G. V. Dunne2 and A. Teplyaev3

1   Department of Applied Physics and Physics, Yale University - New Haven, CT 06520, USA
2   Department of Physics, University of Connecticut - Storrs, CT 06269, USA
3   Department of Mathematics, University of Connecticut - Storrs, CT 06269, USA

received 5 August 2009; accepted in final form 2 November 2009; published November 2009
published online 3 December 2009

It has been realized that fractals may be characterized by complex dimensions, arising from complex poles of the corresponding zeta function, and we show here that these lead to oscillatory behavior in various physical quantities. We identify the physical origin of these complex poles as the exponentially large degeneracy of the iterated eigenvalues of the Laplacian, and discuss applications in quantum mesoscopic systems such as oscillations in the fluctuation $\Sigma ^{2}(E)$ of the number of levels, as a correction to results obtained in random matrix theory. We present explicit expressions for these oscillations for families of diamond fractals, also studied as hierarchical lattices.

05.45.Df - Fractals.
73.23.-b - Electronic transport in mesoscopic systems.
05.60.Gg - Quantum transport.

© EPLA 2009