Eigenvalues of measure theoretic Laplacians on Cantor-like sets

We study the eigenvalues of the Laplacian Δ µ . Here, µ is a singular measure on a bounded interval with an irregular recursive structure, which include self-similar measures as a special case. The structure can also be randomly build. For this operator we determine the asymptotic growth behaviour of the eigenvalue counting function. Furthermore, in the case where µ is self-similar, we give a representation of the eigenvalues of Δ µ as zero points of generalized sine functions allowing, in particular, an explicit computation. Moreover, we use these functions to describe certain properties of the eigenfunctions.