formulate this as a theorem, we want to highlight a further issue related to this problem.\
Above, we have spoken about paths of differential operators and so tacitly assumed continuity. Note that the family has the non-constant domains and so continuity is a non-trivial problem. There are different metrics on spaces of unbounded selfadjoint Fredholm operators on a Hilbert space $H$ and we recommend [@Lesch] for an exhaustive discussion (see also [@Wahl]). A classical approach is to transform unbounded selfadjoint operators $T$ by functional calculus to the bounded selfadjoint operators

$$\begin{aligned}
\label{Riesz}
(I_H+T^2)^{-\frac{1}{2}}\in\mathcal{L}(H),\end{aligned}$$

and to use the operator norm on $\mathcal{L}(H)$ for introducing a distance between unbounded operators. Actually, Atiyah, Patodi and Singer defined the spectral flow in [@APS] for bounded selfadjoint Fredholm operators and applied it to paths of differential operators by using . However, checking continuity along these lines is tedious, if possible at all (see e.g. [@Nicolaescu]), and it seems that the continuity of families of unbounded operators has sometimes been ignored in the literature.\
Every (generally unbounded) selfadjoint operator on a Hilbert space is closed, and there is a canonical metric on the set of all closed operators which is called the gap-metric (see §IV.2 in 