Distributions of localized eigenvalues of Laplacians on post critically finite self-similar sets
The author roughly considers the class of finitely ramified fractals. Assume that such a fractal \(K\) carries a fractal Laplacian \(\Delta\) with a domain \({\mathcal D}_\mu\subset C(K) \subset L^2(K,\mu)\), where \(\mu\) is a finite Borel measure with support \(K\). The so-called post critical set is a natural boundary of \(K\). Under Dirichlet boundary conditions an eigenfunction of \(\Delta\) is called localized if its support is contained in the interior of \(K\). The classical Laplacian on a ball of \(\mathbb{R}^d\) does not have such eigenfunctions but many connected fractals do. The eigenvalue counting function \(\rho(\lambda)\) of \(\Delta\) is decomposed into the sum of a function \(\rho^W(\lambda)\) counting the eigenvalues of localized eigenfunctions and \(\rho^F(\lambda)\) counting the global ones. For \(\lambda \to \infty\) it is shown that \(\rho^W(\lambda)\) asymptotically equals \(\lambda^{d_S/2}\), where \(d_S\) is the spectral dimension of \(K\). On the other hand, some fractals have the asymptotic \(\rho^F(\lambda) \approx \lambda^\beta\) for some \(0<\beta<d_s/2\). In these cases the localized effects dominate the global ones. The techniques are mainly based on those of \textit{J. Kigami} and \textit{M. L. Lapidus} [Commun. Math. Phys. 158, 93-125 (1993; Zbl 0806.35130)].
scientific article; zbMATH DE number 1213991
scientific article