Spectral Analysis on Infinite Sierpiński Gaskets

We study the spectral properties of the Laplacian on infinite Sierpiński gaskets. We prove that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpiński gasket has no boundary, but is false for the Laplacian with the Dirichlet boundary condition. In all these cases we describe the spectrum of the Laplacian and all the eigenfunctions with compact support. To obtain these results, first we prove certain new formulas for eigenprojectors of the Laplacian on finite Sierpiński pre-gaskets. Then we prove that the spectrum of the discrete Laplacian on a Sierpiński lattice is pure point, and the eigenfunctions are localized.