2009/03/31 by Eric Akkermans, E. Akkermans, Gerald V. Dunne +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Degeneracy (biology) #Eigenvalues and eigenvectors #Fractal #Iterated function system #Mesoscopic physics #Physical system #Random Matrices and Applications #Random matrix #Theoretical and Computational Physics #cond-mat.mes-hall #cond-mat.stat-mech #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1209/0295-5075/88/40007
published as Europhys. Lett. 88, 40007 (2009) · 4 pages, 3 figures; v2: references added, as published in Europhysics Letters
openalex publication_date 2009/11/01 · arxiv created 2009/12/04 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Σ 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.