Measure-geometric Laplacians on the real line

Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kreĭn, given an atomless Borel probability measure $\eta$ supported on a compact subset of $\mathbb{R}$, Freiberg and 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 generalise this approach to measures of the form $\eta = \nu + \delta$, where $\nu$ is continuous 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.