Spectral Asymptotics of Generalized Measure Geometric Laplacians on Cantor Like Sets

Abstract

We investigate spectral asymptotic properties of a measure geometric Laplacian which is given as the second derivative w.r.t. two atomless finite Borel measures μ and ν with compact supports L ≔ supp μ and K ≔ supp ν , such that L ⊆ K ⊆ IR. This operator is a generalization of the well-known Sturm-Liouville operator as well as of the measure geometric Laplacian given by . In the special case of self similar measures—Hausdor measures or, more general, self similar measures with arbitrary weights living on Cantor like sets—we determine the asymptotic behaviour of the eigenvalue counting function. This increases under both Dirichlet and Neumann boundary conditions like x γ where the spectral exponent γ is given in dependence of the weights of the measures μ and ν .