Majorana edge states in Kitaev chains of the BDI symmetry class

The paper studies the structure of edge states, but does not address other questions about the description of the different quantum phases, especially in the controversial regime of alpha,beta<1.

The paper deals with a class of interesting models (Kitaev chain with long-range interactions) that, despite their apparent simplicity, displays unusual properties and a rich physics that is yet to be fully understood. The proposed rigorous analytical method is crucial to obtain important results about edge states in this model, which are carefully explained in the manuscript.
The only concern is about a more precise comparison with previous results, such as the one cited in the bibliography (in particular ref. [18]) as well as more recent ones (e.g. Phys. Rev. Lett. 130, 246601 (2022), arXiv:2301.12514). This also implies to enlarge the analysis to other quantities, such as entanglement entropy and/or correlation functions that might shed light on the universal properties of the models. I understand this might be a lot of work, but at least some comments are at order.
The results are interesting and I would suggest publication, after requested changes are taken into account.

Comments about: 1. a stricter comparison with previous results and 2. about the possibility to calculate other physical quantities (e.g. entanglement entropy and/or correlation functions).

1) Clarity and exhaustiveness of the presentation.

2) Novel and relevant technical developments.

I have read the manuscript by the author and I was positively impressed by the in depth investigation of the edge state of the linear chain with different couplings shape. The outlined analytic method appears to be very powerful and i have no reason to doubt the correctness of the calculations. I am inclined to recommend the paper for publication provided that the author is willing to consider a few revision.

The major question mark concerning the paper appears in below Eq. (33) where the author reproduces the result of Ref. [18], but the author expresses some doubts about whether this is the leading behaviour. The author shall make some major comment on what is expected to occur to Eq. (33) upon including further terms in the expansion in Eq. (A10). Indeed, the recent work [arXiv:2301.12514 ] showed how the result in Eq. (33) is not recovered by the scattering approach for $\alpha,\beta>2$, since the leading order momentum terms of the dispersion relation change across this boundary. The author should also acknowledge Ref. [arXiv:2211.15690] which deals on similar matters.

It would be also interesting to comment on how the edge states influences other quantities aside the energy, such as the entanglement entropy of the open hand portion of the chain with respect to the infinite system. In this perspective, please notice that enhanced entanglement scaling is one of the major features of long-range Kitaev chains

10.1088/1742-5468/ac7644

https://doi.org/10.1007/JHEP05(2023)066

1) Comment on the long distance scaling of edge states for \alpha,\beta>2

2) Comment on the possibility to evaluate the entanglement entropy of the system with the method described in the paper.