Physical Consequences of Complex Dimensions of Fractals

Details Physical Consequences of Complex Dimensions of Fractals Physical Consequences of Complex Dimensions of Fractals

2009-03-21

arxiv.org/abs/0903.3681v2

Europhys. Lett. 88, 40007 (2009)

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

4 pages, 3 figures; v2: references added, as published in Europhysics Letters

Scientific paper

It has recently 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.

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