Premium
Localized Eigenfunctions of the Laplacian on p.c.f. Self‐Similar Sets
Author(s) -
Barlow Martin T.,
Kigami Jun
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005358
Subject(s) - eigenfunction , homogeneous space , laplace operator , eigenvalues and eigenvectors , mathematics , fractal , space (punctuation) , function (biology) , lattice (music) , open set , mathematical analysis , class (philosophy) , combinatorics , pure mathematics , mathematical physics , physics , quantum mechanics , geometry , evolutionary biology , linguistics , philosophy , acoustics , biology , artificial intelligence , computer science
In this paper we consider the form of the eigenvalue counting function ρ for Laplacians on p.c.f. self‐similar sets, a class of self‐similar fractal spaces. It is known that on a p.c.f. self‐similar set K the function ρ ( x ) = O ( x d s / 2)as x →∞, for some d s >0. We show that if there exist localized eigenfunctions (that is, a non‐zero eigenfunction which vanishes on some open subset of the space) and K satisfies some additional conditions (‘the lattice case’) then ρ ( x ) x − d s / 2does not converge as x →∞. We next establish a number of sufficient conditions for the existence of a localized eigenfunction in terms of the symmetries of the space K . In particular, we show that any nested fractal with more than two essential fixed points has localized eigenfunctions.