4 - Eigenvalues and Eigenfunctions of Laplacians

Summary

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 ℝⁿ. 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 ℝⁿ. 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.