\,{\mathrm{d}}\mu(y)=0$ since $\mu$ is atomless. Considering $g = f$ we obtain that $\Delta_{\mu,\nu}$ with domain $\mathcal{D}^{*}_{\gamma}( \Delta_{\nu,\mu} )$, for $\gamma\coloneqq (\alpha,\beta) \in [0,\pi/2]^{2}$, is non-positive and since $$\begin{aligned}
    &\int (\Delta_{\nu,\mu}f )gd \nu-\int (\Delta_{\nu,\mu}g )f \mathrm{d} \nu\\
    &=-\tan(\beta)  \nabla_{\mu}f(1) \nabla_{\mu}g(1)-\tan(\alpha)  \nabla_{\mu}f(0) \nabla_{\mu}g(0)+\tan(\beta)  \nabla_{\mu}f(1) \nabla_{\mu}g(1)+\tan(\alpha)  \nabla_{\mu}f(0) \nabla_{\mu}g(0)
    =0,\end{aligned}$$ the boundary conditions force $\Delta_{\mu,\nu}$ also to be symmetric.

Main results
------------

In we establishes a strong connection between $\Delta_{\nu,\mu}$ and $\Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}$. Indeed, by utilising the *pseudo-inverse* $$\begin{aligned}
    \check{F}^{-1}_{\mu}(x) \colon x \mapsto \inf\{ y \in [0,1] \colon F_{\mu}(y) \geq x \}
    \end{aligned}$$ of $F_{\mu}$, we prove, for $* \in \{s,w\}$ and $\gamma \in [0,\pi/2]^{2}$, that $\varphi: f \mapsto f \circ \check{F}_{\mu}^{-1}$ is an isometric isomorphism on $\S^{*}$ with $$\begin{aligned}
  \Delta_{\nu \circ F_{\mu}^{-1}, \Lambda }\circ\varphi=\varphi\circ \Delta_{\nu,\mu }\quad \text{and}\quad\varphi ( \mathcal{D}^*_{\gamma}(\Delta_{\nu,\mu}))=\mathcal{D}^*_{\gamma}(\Delta_{\nu \circ F_{\mu}^{-1},\Lambda}).\end{aligned}$$ With this at hand, one may conclude that the spectral properties of $\Delta_{\nu \circ F_{\mu}^{-1},\Lambda}$ will be inherit from $\Delta_{\nu ,\mu}$ and vice versa. For instance, using our correspondence theorem (), we obtain that $\Delta_{\nu ,\mu}$ is a densely defined, self adjoint operator with compact resolvent. Further, we are able the prove the following.

1.  We obtain concerning the asymtotic growth rate of the eigenvalue counting function of $\Delta_{\nu,\mu}$, first 