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We study the spectral dimensions of Kreȋn-Feller operators for finite Borel measures ν on the d-dimensional unit cube via a form approach. We introduce the notion of the spectral partition function of ν, and assuming that the lower ∞-dimension of ν exceeds d − 2, we show that the upper spectral Neumann dimension coincides with the unique zero of the spectral partition function. We show that if the lower ∞-dimension of ν is strictly less than d − 2, the form approach breaks down. Examples are given for the critical case, that is the lower ∞-dimension of ν equals d − 2, such that for one case the form approach breaks down, another case, where the operator is well defined but we have no discrete set of eigenvalues, and for the third case, where the spectral dimension exists. We provide additional regularity assumptions on the spectral partition function, guaranteeing that the Neumann spectral dimension exists and may coincide with the Dirichlet spectral dimension. We provide examples-namely absolutely continuous measures, Ahlfors-David regular measure, and self-conformal measures with or without overlaps-for which the spectral partition function is essentially given by its L q -spectrum and both the Dirichlet and Neumann spectral dimensions exist. Moreover, we provide general bounds for the upper Neumann spectral dimension in terms of the upper Minkowski dimension of the support of ν and its lower ∞-dimension. Finally, we give an example for which the spectral dimension does not exist. Contents FB 3 -

We study the spectral dimension and asymptotics of Kreȋn-Feller operators for weak Gibbs measures on self-conformal fractals with and without overlaps. We show that the L q -spectrum for every weak Gibbs measure with respect to a C 1 -IFS exists as a limit on the unit interval. Building on recent results of the authors, we can deduce that the spectral dimension with respect to weak Gibbs measures exists and equals the fixed point of the L q -spectrum. For IFS satisfying the open set condition it turns out that the spectral dimension equals the unique zero of the associated pressure function. Moreover, for Gibbs measure with respect to a C 1+γ -IFS under OSC, we are able to determine the asymptotics of the eigenvalue counting function.

We consider generalised Kreȋn-Feller operators ∆ ν,µ with respect to compactly supported Borel probability measures µ and ν under the natural restrictions supp(ν) ⊂ supp(µ) and µ atomless. We show that the solutions of the eigenvalue problem for ∆ ν,µ can be transferred to the corresponding problem for the classical Kreȋn-Feller operator ∆ ν,Λ = ∂ µ ∂ x with respect to the Lebesgue measure Λ via an isometric isomorphism of the underlying Banach spaces. In this way we reprove the spectral asymptotic on the eigenvalue counting function obtained by Freiberg. Additionally, we investigate infinitesimal generators of generalised Liouville Brownian motions associated to generalised Kreȋn-Feller operator ∆ ν,µ under Neumann boundary condition. Extending the measure µ and ν to the real line allows us to determine its walk dimension.