Spectral Asymptotics of Generalized Measure Geometric Laplacians on Cantor Like Sets

Uta Freiberg

Abstract

We investigate spectral asymptotic properties of a measure geometric Laplacian which is given as the second derivative article imagew.r.t. two atomless finite Borel measures μ and ν with compact supports L ≔ supp μ and K ≔ supp ν, such that LK ⊆ IR. This operator is a generalization of the well-known Sturm-Liouville operator article imageas well as of the measure geometric Laplacian given by article image. 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 xγ where the spectral exponent γ is given in dependence of the weights of the measures μ and ν.

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Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

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