Log-Periodic Asymptotic Expansion of the Spectral Partition Function for Self-Similar Sets

Let Z ( t ) be the partition function (the trace of the heat semigroup) of the canonical Laplacian on a post-critically finite self-similar set (with uniform resistance scaling factor and good geometric symmetry) or on a generalized Sierpiński carpet. It is proved that Z ( t ) = ∑ k = 0 n t - d k /...

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Bibliographic Details
Published in:Communications in mathematical physics 2014-06, Vol.328 (3), p.1341-1370
Main Author: Kajino, Naotaka
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
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Summary:Let Z ( t ) be the partition function (the trace of the heat semigroup) of the canonical Laplacian on a post-critically finite self-similar set (with uniform resistance scaling factor and good geometric symmetry) or on a generalized Sierpiński carpet. It is proved that Z ( t ) = ∑ k = 0 n t - d k / d w G k ( - log t ) + O ( exp ( - c t - 1 d w - 1 ) ) as t ↓ 0 for some continuous periodic functions G k : R → R and c ∈ ( 0 , ∞ ) . Here d w ∈ ( 1 , ∞ ) denotes the walk dimension, n  = 1 for a post-critically finite self-similar set, n  =  d for a d -dimensional generalized Sierpiński carpet, { d k } k = 0 n ⊂ [ 0 , ∞ ) is strictly decreasing with d n  = 0,  G 0 is strictly positive and G 1 is either strictly positive or strictly negative depending on the (Neumann or Dirichlet) boundary condition.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-014-1922-3