[@FZ02] and, when $\mu$ is a self-similar measure supported on a Cantor set, it is now well established that the eigenvalue counting function of $\Delta_{\mu, \mu}$ is comparable to the square-root function. In [@KSW16] the exact eigenvalues and eigenfunctions of $\Delta_{\mu, \mu}$ were obtained and it was shown that the eigenvalues do not depend on the given measure. Moreover, the eigenfunctions are given by a composition of the appropriated classical trigonometric functions with a phase space transformation induced by the distribution function of $\mu$. The case when the measure $\mu$ has a continuous as well as an atomic part, was the subject of [@KSW17; @KSW2019b]. Here, it has been shown that, if $\mu$ has a continuous part, then the eigenvalues may depend on the position of the atoms, and otherwise not.

In the present article, we elaborate on the connections between the generalised and the classical Kreĭn-Feller operators by establishing a suitable phase space transformation determined by the distribution function of $\mu$. As a first application of this observation we show that the spectral properties of the generalised Kreĭn-Feller operators can be reduced to those of the to the classical ones; and as a second application, we connect properties of 