Uta Renata Freiberg

Generalized second order differential operators of the form $ {d \over {d \mu}} {d \over {dx}} $ when μ is a selfsimilar measure whose support is the classical Cantor set are considered. The asymptotic distribution of the zeros of the eigenfunctions is determined.

We investigate analytical properties of a measure geometric Laplacian which is given as the second derivative $ {d \over {d \mu}} {d \over {d \nu}} $ w.r.t. two atomless finite Borel measures μ and ν with compact supports supp μ ⊂ supp ν on the real line. This class of operators includes a generalization of the well‐known Sturm‐Liouville operator $ {d \over {d \mu}} {d \over {dx}} $ as well as of the measure geometric Laplacian given by $ {d \over {d \mu}} {d \over {d \mu}} $. We obtain for this differential operator under both Dirichlet and Neumann boundary conditions similar properties as known in the classical Lebesgue case for the euclidean Laplacian like Gauß‐Green‐formula, inversion formula, compactness of the resolvent and its kernel representation w.r.t. the corresponding Green function. Finally we prove nuclearity of the resolvent and give two representations of its trace.