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](https://doi.org/10.1002/cpa.21812)" Communications on Pure and Applied Mathematics 72.5 (2019): 1044-1119, [MR3935478](https://mathscinet.ams.org/mathscinet-getitem?mr=MR3935478), [Zbl 1418.35190](https://zbmath.org/?q=an%3A1418.35190).

[2] Rupert L. Frank and Enno Lenzmann, "[Uniqueness of non-linear ground states for fractional Laplacians in $\Bbb R$](https://doi.org/10.1007/s11511-013-0095-9)" Acta mathematica 210.2 (2013): 261-318, [MR3070568](https://mathscinet.ams.org/mathscinet-getitem?mr=MR3070568), [Zbl 1307.35315](https://zbmath.org/?q=an%3A1307.35315).