The Faddeev-Merkuriev Differential Equations (MFE) and Multichannel 3-Body Scattering Systems

Round  1

Reviewer 1 Report

The presented manuscript is a supplement of the previously published research by the same author [23]. Nevertheless, in view of the complexity of the performed calculations and for the first time realised rigorous numerical exploration of the intriguing phenomena predicted some fourty years ago (M. Gailitis and R. Damburg, Journal of Experimental and Theoretical Physics, Vol. 17, 1963, p. 1107) this manuscript deserves being published in some form. However I will expect in a revised version stronger effort done in explaining physical origin of this phenomenon. Here is a list of my suggestions/remarks:

1) It would be useful to precise energy region one explores in this paper. I.e. inbetween P+ps(n=2) and e^+ +H(n=3) thresholds.

2) I am not convinced if the considered increase in the cross-section is 'resonance phenomena' related with S-matrix poles in complex energy plane above the P+ps(n=2) threshold and the resulting Breit-Wigner distribution

a) If these are result of resonant S-matrix poles, why positions of these resonances are not established in numerous calculations of the resonances (like Complex Scaling method, used by the same author some time ago)

b) As I could understand from the seminal paper of Gailitis and Damburg it is suggested that the effect is due to the presence of the well-konwn Feshback resonances below the P+ps(n=2) threshold and a consequence of Levinsons theorem, requiring n\pi shift in the phaseshifts.

I believe by manipulating results presented in this paper this question might be clearly answered.

3) I could not understand why the author abandoned the method which takes into account modification of the boundary condition due to the presence of 1/y^2 potential. Even though using large grids one may limit effect of the modified boundary condition, its proper consideration should be strongly benificial

4) Clearly seminal work of M. Gailitis and R. Damburg should be cited!

Author Response

Dear referee:

I would like to thank you for reading this manuscript. Indeed, this is the first calculation of its kind. I am afraid that this might become the last calculation of its kind! So the purpose of this note, as described in the abstract, is to help the students in their development and testing of their own MFE codes.

We have just published another article[25] which can explain more clearly all your questions. Please find it in this link: https://www.mdpi.com/2218-2004/4/1/8/html

Your kindness is gratefully appreciated.

c. y. Hu

Reviewer 2 Report

This is an interseting paper for the few-body and positron atomic physics community. The author use a very well known few-body techniques

based on the three-body Faddeev-Merkuriev theory. In my opinion, this work can be accepted for publication in the current form.

Author Response

Dear Referee #3

Thank you so very much for your kind encouragement.

Best Regards,

C. Y. Hu

Reviewer 3 Report

During the last years the special asymptotic behaviour of the three-body wave function for the case of three charged particles (for the special configurations when two particles are close to each other and the third one is far away) has been discussed in the literature:

V.S.Buslaev, S.B.Levin, "A system of three three-dimensional charged quantum particles: Asymptotic behaviour of the eigenfunctions of the continuous spectrum at infinity",

Functional analysis and its applications, 46,2, pp147-151, (2012).

Do you think it coud be important for the accuracy of your calculations?

Author Response

Dear referee:

Thank you for reading this manuscript and your valuable suggestion concerning the asymptotic wave function. Unfortunately, we did the calculation a few years ago on the supercomputer Ranger. Ranger no longer exists. It is not so simple to adopt the code for another supercomputer! In [23],we solved a half million coupled linear equations. we have been very careful to ensure the stability and symmetry of the 6 X 6 K-matrix.

The cutoff distance is three orders of magnitude larger than the target system. We were interested only in the six channel wave amplitudes,

did not calculate the total wave function. However,we found very interesting physics in the channel wave amplitudes, oppose from common belief! , that motivates me to submit this manuscript to encourage students in their development and testing their own MFE code.

I believe a few more efficient MFE codes would benefit all physical sciences and atomic physics in particular.

Thank you for your patience,

C.Y. Hu

Round  2

Reviewer 1 Report

The presented manuscript is a supplement of the previously published research by the same author [23]. Nevertheless, in view of the complexity of the performed calculations this manuscript deserves being published.