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Keywords:
Kreĭn--Feller operator; spectral asymptotics; harmonic analysis
Summary:
Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M.\,G. Kreĭn, given an atomless Borel probability measure $\eta$ supported on a compact subset of $\mathbb R$ U. Freiberg and M. Zähle 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.
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