supported Borel probability measure on $\mathbb{R}$ and where $\Lambda$ denotes the one-dimensional Lebesgue measure, has been investigated with respect to its spectral properties first by Fujita [@Fu87], Küchler [@MR574035], Langer [@MR0314125] and Kotani and Watanabe [@MR661628] and more recently by Arzt [@A15b], Ehnes [@Ehnes2019] and Freiberg [@MR2017701; @MR2030736; @Fr05]. The case when $\nu$ is purely atomic has also been studied in [@MR2513598]; where it was shown that the eigenvalues of $\Delta_{\nu, \Lambda}$ have a dependence not only on the positions of the atoms of $\nu$ but also on the weights of the atoms. Returning to the case when $\nu$ is a non-atomic, it has been established that $\Delta_{\nu, \Lambda}$ is the infinitesimal generator of a Liouville Brownian motion (also known as gap diffusion, skip-free diffusion, quasi-diffusion or generalised diffusion), see [@Burkhardt1983; @MR2817342; @MR3034785; @X_Jin_2017; @MR574035; @MR0314125; @MR3005002; @MR3272822].

Here, we investigate generalised Kreĭn-Feller operators $\Delta_{\nu, \mu}$ for Borel measures $\nu$ and $\mu $ on the real line under the natural assumptions ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}( \mu)$ and $\mu$ atomless. In the case that $\nu=\mu =\Lambda$ the operator coincides with the classical second order weak derivative. For arbitrary $\mu=\nu$, atomless and compactly supported, a harmonic calculus for $\Delta_{\mu, \mu}$ was developed in 