very convenient to obtain paths in $\Lambda(n)$ with a given Maslov index.\
Let us first consider the path

$$[0,1]\ni \lambda\mapsto A(\lambda)=\operatorname{diag}(e^{i\pi\lambda},1,\ldots,1)\in U(n)$$ and its projection $\overline{A}(\lambda):=A(\lambda)\cdot O(n)$ to the quotient $U(n)/O(n)$. Note that$A(0)\operatorname{diag}(-1,1,\ldots,1)=A(1)$ and so $\overline{A}$ is a closed curve. Also, as $\det^2(A(\lambda))=e^{2\pi i\lambda}$, we see that the Maslov index of the corresponding path in $\Lambda(n)$ is $1$. Using the identification $\mathbb{C}^n\cong\mathbb{R}^{2n}$, it is readily seen that

$$\gamma_{nor}(\lambda):=A(\lambda)(\mathbb{R}^n\times\{0\})=\mathbb{R}(\cos(\pi\lambda)e_1+\sin(\pi\lambda)e_{n+1})+\sum^{n}_{j=2}{\mathbb{R}e_j}\in\Lambda(n).$$ Hence we have found a path $\gamma_{nor}$ such that $\gamma_{nor}(0)=\gamma_{nor}(1)=\mathbb{R}^n\times\{0\}$ and $\mu_{Mas}(\gamma_{nor})=1$.\
Let us now consider

$$[0,1]\ni \lambda\mapsto B(\lambda)=\operatorname{diag}(-ie^{ i\pi\lambda},i,\ldots,i)\in U(n)$$ and note that again the projection $\overline{B}$ to $U(n)/O(n)$ is a closed path and $\det^2(B(\lambda))=(-1)^{n}e^{2\pi i\lambda}$. Hence

$$\gamma'_{nor}(\lambda):=B(\lambda)(\mathbb{R}^n\times\{0\})=\mathbb{R}(\sin(\pi\lambda)e_1-\cos(\pi\lambda)e_{n+1})+\sum^{2n}_{j=n+2}{\mathbb{R}e_j}\in\Lambda(n)$$ is such that $\gamma'_{nor}(0)=\gamma'_{nor}(1)=\{0\}\times\mathbb{R}^n$ and $\mu_{Mas}(\gamma'_{nor})=1$.

The Gap-Metric and the Spectral Flow {#section-spectralflow}
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Our first aim of this section is to recall the definition of the gap-metric, where we follow Kato’s monograph [@Kato].\
Let $H$ be a real Hilbert space and let $G(H)$ denote the set of all closed subspaces of $H$. For every $U\in G(H)$ there is a unique orthogonal projection $P_U$ onto $U$ which is a bounded operator on $H$. We set

$$d_G(U,V)=\|P_U-P_V\|,\quad U,V\in G(H),$$ and note that this is obviously a metric on $G(H)$. The distance between two non-trivial 