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

TL;DR

This paper explores how eigenfunctions of the Neumann Laplacian vary continuously as the domain shape changes, including explicit examples like squares to circles and approximations of the Sierpinski carpet.

Contribution

It introduces the concept of homotopies of eigenfunctions and provides explicit examples, including the novel case of the Sierpinski carpet approximation.

Findings

Eigenfunctions form continuous families during domain deformations.

Explicit homotopies connect square and circle eigenfunctions.

Approximation of the Sierpinski carpet demonstrates complex spectral behavior.

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)