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LOCALIZED EIGENFUNCTIONS OF THE LAPLACIAN ON p.c.f. SELF-SIMILAR SETS

Published online by Cambridge University Press:  01 October 1997

MARTIN T. BARLOW  and
JUN KIGAMI
MARTIN T. BARLOW
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada. E-mail address; [email protected]
JUN KIGAMI
Affiliation:
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-01, Japan. E-mail address: [email protected]
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Abstract

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(xds/2) as x→∞, for some ds>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−ds/2 does 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.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 320 - 332
Copyright
The London Mathematical Society 1997

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Footnotes

This research was partly supported by an NSERC of Canada grant.