Theory of Quantum Mechanical Scattering in Hyperbolic Space
, Y. A. Kurochkin 2,*, V. S. Otchik 3, P. F. Pista 4, N. D. Shaikovskaya 2 and D. V. Shoukavy 2Round 1
Reviewer 1 Report
This article combines previous articles by the same authors, some old, like: ON THE QUANTUM-MECHANICAL SCATTERING PROBLEM
IN THE LOBACHEVSKY SPACE, Rom.J. Phys 50, 2005, pp. 37–44, and some new: Yu. A. Kurochkin, V.S. Otchik, N.D. Shaikovskaya and Dz. V. Shoukavy. Quantum-Mechanical Scattering Problem in Lobachevsky Space at Low Energies. Nonlinear Phenomena in Complex Systems, 25, no. 3, 2022, p. 245 – 253. In particular, the last one has almost the same name and abstract as the present article. The only change is Lobachevsky -> Hyperbolic. So unless the authors make it very clear in their introduction and along the text what their new results are, the article is unsuitable for publication in Symmetry.
Author Response
The theory of scattering in Lobachevsky (hyperbolic) space is as extensive as in the three-dimensional flat space. In an article published in the journal Rom.J. Phys 50, 2005, pp. 37–44 gives the general solution for the scattering problem on the spherical symmetrical potential well. It should be emphasized that the general solution is cumbersome and therefore not very convenient for application. Thus, in the work Nonlinear Phenomena in Complex Systems, 25, no. 3, 2022, p. 245 – 253, the case low-energy scattering for short-range potentials is considered. We guess that it has prospects for use in nuclear physics.
In submitted manuscript expressions for scattering lengths in the Coulomb problem in flat space (see Eq, 34) as well as hyperbolic space (see Eq. 40) are obtained for the first time. The scattering length in flat space as follows from Eq. 24 does not contain the parameters of the Coulomb potential that looks a bit strange at first. The obtained the same formula for hyperbolic space (Eq. 40) confirms the correctness of formula (34).
It should be note that even in the fundamental monograph Quantum Mechanics by L. Landau and E. Lifshitz there is no formula for the scattering length for the Coulomb problem. In the monograph only notes that this value is large.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors analyze scattering problem in quantum mechanics on the hyperbolic space.
General theory of scattering goes similarly to the case of the Euclidean space, where the scattering length can also be defined on the hyperbolic space.
The cases of the spherically symmetric potential well and the Coulomb potential are considered.
In the former, the minimum energy for the scattering depends on the curvature radius and does not vanish as long as the curvature radius is finite. For the case that the energy lower than the minimum, only bound states are possible. In the latter, the scattering length remains finite for the low energy limit, which means that the Coulomb force on the hyperbolic space is not of long range, differently from the case on the Euclidean space.
As I think that this paper is interesting, I can recommend the publication after some minor typos as given below.
Abstract, line 4
of the of the scattering -> of the scattering
Page 1
line 2 of the paragraph ``In a series of papers ...'': similar to the of scattering -> similar to the scattering
Page 6
eq.(38): I guess that the factor e^{i(\gamma_+ - i\gamma_-)} should be changed to ^{i(\gamma_+ - \gamma_-)}.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
In the Equations (4), (6) in not defined the notation P^{-1/2-l}_{-1/2+i\eta}(ch \beta). A full definition must be included or a reference.
Including this reference or definition the article could be published in Symmetry.
Author Response
Response to Reviewer 1 Comments (Round 2)
Point 1: In the Equations (4), (6) in not defined the notation P^{-1/2-l}_{-1/2+i\eta}(ch \beta). A full definition must be included or a reference.
Response 1: The full definition of the notation P^{-1/2-l}_{-1/2+i\eta}(ch \beta) is included at the manuscript, after Eq. (4) as
\added{where $P^{-\frac{1}{2}-l}_{-\frac{1}{2}+i \eta}(ch\beta)$ -- Legendre functions of the first kind.}.
Please see the attached the latest version of manuscript.
Author Response File: Author Response.pdf
