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Exactly solvable quantum fewbody systems associated with the symmetries of the threedimensional and fourdimensional icosahedra
by T. Scoquart, J. J. Seaward, S. G. Jackson, M. Olshanii
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Submission summary
Authors (as registered SciPost users):  Maxim Olshanii 
Submission information  

Preprint Link:  http://arxiv.org/abs/1608.04402v2 (pdf) 
Date submitted:  20160927 02:00 
Submitted by:  Olshanii, Maxim 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The purpose of this article is to demonstrate that noncrystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a oneparametric family of solvable fourbody systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a halfline. We repeat the program for a 600cell, a fourdimensional generalization of the regular threedimensional icosahedron.
Current status:
List of changes
1. Added a reference to Heckman and Opdam's "Yang’s system of particles and Hecke algebras."
2. Added a second paragraph to the concluding section about extending the results to finitestrength deltapotentials.
3. Minor grammatical, spelling and wordchoice changes.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2016103 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1608.04402v2, delivered 20161003, doi: 10.21468/SciPost.Report.26
Strengths
1. Finding specific mass ratios in the fewbody hardcore models with particleparticle contact interaction for which one obtains H_3 and H_4 type LiebLiniger models with infinite delta interaction strenghts by a change of coordinates.
2. Obtaining an explicit wave expansion of the solutions of the associated spectral problem.
Weaknesses
1. Identification of LiebLiniger models associated to exceptional reflection groups and fewbody models
can only be done for infinite deltainteraction strengths (hardcore particles). A precise analysis what prevents extension to finite deltainteraction strengths is missing.
2. The ideas in this paper have been worked out before for reflection group of type F_4 in ref. [4] of the paper under review. The paper is a rather straightforward exercise to adjust the techniques to reflection
groups of type H_3 and H_4.
Report
The LiebLiniger Bose gas on the line is a famous integrable onedimensional many body system naturally attached to the symmetric group. It admits a generalisation as integrable system in which the role of the symmetric group is taken over by a finite reflection group. The particleparticle contact interaction of the LiebLiniger Bose gas is replaced by deltainteractions at the root hyperplanes of the reflection group.
For classical Weyl groups there still is a reasonable interpretation as a onedimensional many body system. For other types, in case of infinite deltainteraction strengths, the model can sometimes admit an interpretation as a hardcore fewbody model on the line with distinguishable particles.
The paper under review gives an example of this phenomenon.
The idea that onedimensional particles with different masses and with particleparticle contact interaction can be related by a (nonorthogonal) change of coordinates to a model describing equal particles but with deltapotential interactions along "nonphysical" hyperplanes goes back to McGuire in 1963 (it is reference [7] in the paper under review). The main point of the paper is to show that if the masses of the particles are exactly such that the hyperplanes x_i=x_{i+1} turn into the simple root hyperplanes for some nonclassical finite reflection group, then the model is equivalent to the associated generalised LiebLiniger model with infinite deltainteraction strengths. In ref. [4] of the paper under review this was already worked out in case of the finite reflection group of type F_4 and its affine version. In the present paper the same techniques are used for the noncrystallographic reflection groups H_3 and H_4.
Requested changes
1. See item 1 at weaknesses. A thorough analysis of what goes wrong for finite interaction strengths should be added.
Small comments:
2. plains > planes (several times).
3. Section 2: Formula for e_{COM}\cdot\mathbf{z} is wrong (only so for \mu=M).
4. "m_{i1}m_i and m_im_{i+1} planes" need explanation.
5. I do not understand why giving an arctanformula for the angle, instead of the (more standard) way of expressing the angle using arccos (which does not make the massdependence more difficult).
6. Page 3: Ref. [15] should be [16] I suppose.
Author: Maxim Olshanii on 20161009 [id 59]
(in reply to Report 1 on 20161003)We thank the referee for their careful reading and insightful critique.
>> 1. See item 1 at weaknesses. A thorough analysis of what goes wrong for finite
>>interaction strengths should be added.
We added an explanatory paragraph in the conclusion section.
>> Small comments:
>> 2. plains > planes (several times).
Corrected
>> 3. Section 2: Formula for e_{COM}\cdot\mathbf{z} is wrong (only so for \mu=M).
Corrected
>> 4. "m_{i1}m_i and m_im_{i+1} planes" need explanation.
Corrected
>> 5. I do not understand why giving an arctanformula for the angle, instead of the (more >> standard) way of expressing the angle using arccos (which does not make the mass
>> dependence more difficult).
The convention is consistent with the well accepted convention used in the
foundational McGuire article and the rest of our work in this topic.
>> 6. Page 3: Ref. [15] should be [16] I suppose.
Corrected