On eigenvalue problems for Laplacians on p.c.f. self-similar sets
type: Item
instance of: scholarly article
title: On eigenvalue problems for Laplacians on p.c.f. self-similar sets
zbMATH Open document ID: 0861.58047
DOI: 10.1007/BF03167295
author: Tadashi Shima
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 1997-05-12T00:00:00Z
review text: This paper considers a strong harmonic structure under which eigenvalues of the Laplacian on a p.c.f. self-similar set are completely determined according to the dynamical system generated by a rational function. The author shows that, with some additional assumptions, the eigenvalue counting function \(\rho(\lambda)\) behaves so wildly that \(\rho(\lambda)\) does not vary regularly, and the ratio \(\rho(\lambda)/\lambda^{d_s/2}\) is bounded but not convergent as \(\lambda \nearrow \infty\), where \(d_s\) is the spectral dimension of the p.c.f. self-similar set.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 37A30
zbMATH DE Number: 927306
zbMATH Keywords: eigenvalue
zbMATH Keywords: post critically finite self-similar sets
zbMATH Keywords: spectral decimation
zbMATH Keywords: spectral dimension
zbMATH Keywords: Laplacian
reviewed by: Stana Nikčević
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 927306