Physical Consequences of Complex Dimensions of Fractals

TL;DR

This paper explores how complex dimensions of fractals, derived from their zeta functions, induce oscillations in physical quantities, with applications in quantum systems and explicit formulas for diamond fractals.

Contribution

It demonstrates that complex poles of fractal zeta functions cause oscillatory behavior in physical measures and links these poles to eigenvalue degeneracies in quantum systems.

Findings

Complex poles lead to oscillations in physical quantities.

Explicit formulas for oscillations in diamond fractals.

Application to quantum mesoscopic systems and random matrix theory.

Abstract

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.