introduced by Cappell, Lee and Miller in [@Cappell]. Let us first recall that the spectral flow is a homotopy invariant for paths of selfadjoint Fredholm operators that was invented by Atiyah, Patodi and Singer in [@APS], and since then has been used in various different settings (see e.g. [@Fredholm §5.2]). The spectrum of a selfadjoint Fredholm operator consists only of eigenvalues of finite multiplicity in a neighbourhood of $0\in\mathbb{R}$ and, roughly speaking, the spectral flow of a path of such operators is the net number of eigenvalues crossing $0$ whilst the parameter of the path travels along the interval.\
Let us now consider for a pair of paths $(\gamma_1,\gamma_2)$ in $\Lambda(n)$ the differential operators

$$\begin{aligned}
\label{ops-def}
\mathcal{A}_\lambda:\mathcal{D}(\mathcal{A}_\lambda)\subset L^2(I,\mathbb{R}^{2n})\rightarrow L^2(I,\mathbb{R}^{2n}),\quad (\mathcal{A}_\lambda u)(t)=Ju'(t),\end{aligned}$$

where

$$\begin{aligned}
\label{ops-def-domain}
\mathcal{D}(\mathcal{A}_\lambda)=\{u\in H^1(I,\mathbb{R}^{2n}):\, u(0)\in\gamma_1(\lambda), u(1)\in\gamma_2(\lambda)\}.\end{aligned}$$

By an elementary computation, $\mathcal{A}_\lambda$ is symmetric, and it is also not difficult to see that it actually is a selfadjoint Fredholm operator. Note that the kernel of $\mathcal{A}_\lambda$ is isomorphic to $\gamma_1(\lambda)\cap\gamma_2(\lambda)$, which suggests that the spectral flow of the path $\mathcal{A}=\{\mathcal{A}_\lambda\}_{\lambda\in I}$ is related to the Maslov index of the pair $(\gamma_1,\gamma_2)$. As we already mentioned above, their equality is one of the main achievements of [@Cappell]. However, before we 