Generalised Kreĭn–Feller operators and gap diffusions via transformations of measure spaces

We consider the generalised Krein-Feller operator Δν,μ with respect to compactly supported Borel probability measures μ and ν with the natural restrictions that μ is atomless, the supp(ν)⊆supp(μ) and the atoms of ν are embedded in the supp(μ). We show that the solutions of the eigenvalue problem for Δν,μ can be transferred to the corresponding problem for the classical Krein-Feller operator Δν∘F−1μ,Λ with respect to the Lebesgue measure Λ via an isometric isomorphism determined by the distribution function Fμ of μ. In this way, we obtain a new characterisation of the upper spectral dimension and consolidate many known results on the spectral asymptotics of Krein-Feller operators. We also recover known properties of and connections to generalised gap diffusions associated to these operators.