We consider the generalised Kre\u{\i}n-Feller operator Δν,μ
with respect to compactly supported Borel probability measures μ and ν
under the natural restrictions supp(ν)⊆supp(μ)
and μ is atomless. We show that the solutions of the eigenvalue problem for
Δν,μ can be transferred to the corresponding problem for the
classical Kre\u{\i}n-Feller operator Δν,Λ=∂μ∂x with respect to the Lebesgue measure
Λ via an isometric isomorphism of the underlying Banach spaces. In this
way we reprove and consolidate many known results on the spectral asymptotics
of Kre\u{\i}n-Feller operators. Additionally, we investigate infinitesimal
generators of generalised gap diffusions associated to generalised
Kre\u{\i}n-Feller operators under Neumann boundary condition and determine
their scale functions and speed measures. Extending the measure μ and ν
to the real line allows us to determine the walk dimension of the given gap
diffusion.Comment: 15 pages, 4 figure