Abstract:Motivated by the fundamental theorem of calculus, and based on the works of W.~Feller as well as M.\ Kac and M.\,G.\ Kre\u\i n, given an atomless Borel probability measure $\eta$ supported on a compact subset of $\mathbb R$ U.\ Freiberg and M.\ Z\"ahle introduced a measure-geometric approach to define a first order differential operator $\nabla_{\eta}$ and a second order differential operator~$\Delta_{\eta}$, with respect to $\eta$. We generalize this approach to measures of the form $\eta := \nu + \delta$, where $\nu$ is non-atomic and $\delta$ is finitely supported. We determine analytic properties of $\nabla_{\eta}$ and $\Delta_{\eta}$ and show that $\Delta_{\eta}$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $\Delta_{\eta}$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.
Keywords: Kre\u\i n--Feller operator; spectral asymptotics; harmonic analysis
DOI: DOI 10.14712/1213-7243.2020.026
AMS Subject Classification: 47G30 42B35 35P20