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

Consider a family of bounded domains Ωt in the plane (or more generally any Euclidean space) that depend analytically on the parameter t, and consider the ordinary Neumann Laplacian ∆t on each of them. Then we can organize all the eigenfunctions into continuous families u (j) t with eigenvalues λ (j) t also varying continuously with t, although the relative sizes of the eigenvalues will change with t at crossings where λ (j) t = λ (k) t . We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has Ω0 equal to a square and Ω1 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)