Distributions of Localized Eigenvalues of Laplacians on Post Critically Finite Self-Similar Sets

In this paper, we study distributions of eigenvalues corresponding to localized eigenfunctions of Laplacians on p.c.f. self-similar sets. Precisely, we divide the eigenvalue counting functionρ(x) of a Laplacian into two parts,ρW(x) andρF(x), whereρW(x) is the counting function of localized eigenvalues andρF(x) is the counting function of non-localized (global) eigenvalues. We study asymptotic behaviors ofρW(x) andρF(x) asx→∞. It is shown thatρW(x)≈xdS/2wheredSis the spectral exponent. On the other hand, for a class of Laplacians, including the standard Laplacian on the Sierpinski gasket,ρF(x)≈xκFfor someκF<dS/2. So localized eigenfunctions dominate global eigenfunctions in such cases.