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)

54 pages, 37 figures, 2 tables, to appear, Fractals

References in corpus (1)

• Outer Approximation of the Spectrum of a Fractal Laplacian