Eigenvalues and Eigenfunctions of Laplacians
Jun Kigami · Cambridge University Press eBooks · 2001
In this chapter, we will study eigenvalues and eigenfunctions for the Laplacian Δμ associated with ( D , r) and μ. In particular, we will be interested in the asymptotic behavior of the eigenvalue counting function and present a Weyl-type result (Theorem 4.1.5) in 4.1. It turns out that the nature of eigenvalues and eigenfunctions of Δμ is quite different from that of Laplacians on a bounded domain of ℝ n . For example, we will find localized eigenfunctions in certain cases. More precisely, in 4.3, we will define the notion of pre-localized eigenfunctions, which are the eigenfunction of Δμ satisfying both Neumann and Dirichlet boundary conditions. It is known that such an eigenfunction does not exists for the ordinary Laplacian on a bounded domain of ℝ n . Proposition 4.3.3 shows that if there exists a pre-localized eigenfunction, then, for any open set O ⊆ K , there exists a pre-localized eigenfunction whose support is contained in O . One important consequence of the existence of pre-localized eigenfunctions is the discontinuity of the integrated density of states. See Theorem 4.3.4 and the remark after it. We will give a sufficient condition for the existence of pre-localized eigenfunctions in 4.4. In particular, we will see that there exists a pre-localized eigenfunction for the Laplacian on an affine nested fractal associated with the harmonic structure appearing in Theorem 3.8.10. See Corollary 4.4.11.