I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g. [1] and [2]). However the definition differs for each context. As far as I know this came from Morse theory in differential topology. But I am not sure how the definition of "Morse index" in [2] can be related to the definition in the differential topology. The definition in [2] is:

For a given (fractional) linear differential operator $L_+$, Morse index of $L_+$ in the sector of even functions is defined by $$\mathcal{N}_{-,\text{even}}(L_+):=\#\{e<0:e\text{ is eigenvalue of }L_+\text{ restricted to }L_\text{even}^2(\mathbb{R})\}.$$

As this seems functional analysis terminology, how does this relate to the definition in the differential topology, which essentially is the number of critical points in a given manifold? Negative eigenvalue of a linear differential operator somewhat resembles critical points in a manifold?

Sorry for my ignorance, I am new to this topic, and thank you in advance.

References

[1] Kelei Wang and Juncheng Wei, "Finite Morse index implies finite ends" Communications on Pure and Applied Mathematics 72.5 (2019): 1044-1119, MR3935478, Zbl 1418.35190.

[2] Rupert L. Frank and Enno Lenzmann, "Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb R$" Acta mathematica 210.2 (2013): 261-318, MR3070568, Zbl 1307.35315.