  Motivated by the fundamental theorem of calculus, and based on the works of
Feller as well as Kac and Kre\u{\i}n, given an atomless Borel probability
measure $\eta$ supported on a compact subset of $\mathbb{R}$, Freiberg and
Z\"{a}hle 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.
