Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet

Abstract

Consider a family of bounded domains $\Omega_{t}$ in the plane (or more generally any Euclidean space) that depend analytically on the parameter $t$, and consider the ordinary Neumann Laplacian $\Delta_{t}$ on each of them. Then we can organize all the eigenfunctions into continuous families $u_{t}^{(j)}$ with eigenvalues $\lambda_{t}^{(j)}$ also varying continuously with $t$, although the relative sizes of the eigenvalues will change with $t$ at crossings where $\lambda_{t}^{(j)}=\lambda_{t}^{(k)}$. We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has $\Omega_{0}$ equal to a square and $\Omega_{1}$ equal to a circle; in both cases the eigenfunctions are known explicitly, so our homotopies connect these two explicit families. In the second example we approximate the Sierpinski carpet starting with a square, and we continuously delete subsquares of varying sizes. (Data available in full at www.math.cornell.edu/~smh82)