Exact spectrum of the Laplacian on a domain in the Sierpinski gasket

Abstract

For a certain domain $\Omega$ in the Sierpinski gasket $\mathcal{SG}$ whose boundary is a line segment, a complete description of the eigenvalues of the Laplacian, with an exact count of dimensions of eigenspaces, under the Dirichlet and Neumann boundary conditions is presented. The method developed in this paper is a weak version of the spectral decimation method due to Fukushima and Shima, since for a lot of "bad" eigenvalues the spectral decimation method can not be used directly. Let $\rho^0(x)$, $\rho^\Omega(x)$ be the eigenvalue counting functions of the Laplacian associated to $\mathcal{SG}$ and $\Omega$ respectively. We prove a comparison between $\rho^0(x)$ and $\rho^\Omega(x)$ says that $ 0\leq \rho^{0}(x)-\rho^\Omega(x)\leq C x^{\log2/\log 5}\log x$ for sufficiently large $x$ for some positive constant $C$. As a consequence, $\rho^\Omega(x)=g(\log x)x^{\log 3/\log 5}+O(x^{\log2/\log5}\log x)$ as $x\rightarrow\infty$, for some (right-continuous discontinuous) $\log 5$-periodic function $g:\mathbb{R}\rightarrow \mathbb{R}$ with $0<\inf_{\mathbb{R}}g<\sup_\mathbb{R}g<\infty$. Moreover, we explain that the asymptotic expansion of $\rho^\Omega(x)$ should admit a second term of the order $\log2/\log 5$, that becomes apparent from the experimental data. This is very analogous to the conjectures of Weyl and Berry.