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Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra
by T. Scoquart, J. J. Seaward, S. G. Jackson, M. Olshanii
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|Authors (as Contributors):||Maxim Olshanii|
|Arxiv Link:||http://arxiv.org/abs/1608.04402v1 (pdf)|
|Date submitted:||2016-08-17 02:00|
|Submitted by:||Olshanii, Maxim|
|Submitted to:||SciPost Physics|
The purpose of this article is to demonstrate that non-crystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a one-parametric family of solvable four-body systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a half-line. We repeat the program for a 600-cell, a four-dimensional generalization of the regular three-dimensional icosahedron.
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Reports on this Submission
Anonymous Report 1 on 2016-9-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1608.04402v1, delivered 2016-09-13, doi: 10.21468/SciPost.Report.15
1- An new way of looking at Guadin's system associated with the finite Coxeter groups H_3, H_4 in the *impenetrable* regime, namely as few-body particle systems with different masses.
1- Only hard-core particles are considered. It is a natural question whether Gaudin's system  associated with H_3 and H_4 in the *interacting* regime can also be interpreted as a few-body system.
Gaudin observed in a landmark paper that the quantum boson system of $n$ particles with delta-potential interaction admits natural generalisation in terms of finite Coxeter groups . Furthermore, the generalised systems from  also admits generalisations in terms of affine Weyl groups [1,2, 8,9,10,11,12,13].
For classical Coxeter groups these systems have natural particle interpretations. For example, the original n-particle model with pairwise interaction corresponds to the permutation group $S_n$ (on the line) or affine $S_n$ (on the circle). The other classical Coxeter groups appear when restricting the particles to a half-line or interval, or by distributing them symmetrically around the origin.
Recently the last two authors showed that the system associated with the affine Weyl group $\tilde F_4$ in the impenetrable regime (i.e. when the interaction goes to +infinity) can be mathematically identified with a system of four one-dimensional hard-core particles with mass ratios 6, 2, 1, and 3 in a hard-wall box.
The preprint under review is a continuation of the above idea, but now applied to the finite non-crystallographic Coxeter groups of type H_3 and H_4. The paper can be summarised as follows: In the impenetrable regime Gaudin's systems  associated with H_3 and H_4 can be mathematically identified with a system of four (for H_3) or five (for H_4) one-dimensional hard-core particles on a line, if the ratios of the masses m_j/m_2 are certain explicit rational functions of a dimensionless parameter \xi. For very special values of $\xi$ the system reduces to a system of three (for H_3) or four (for H_4) one-dimensional hard-core particles on a half-line.
1- In the paper "Yang's system of particles and Hecke algebras", Heckman and Opdam relate the spectral problem for the system in  to the representation theory of the graded Hecke algebra of the corresponding Coxeter group. Furthermore, they also prove the crucial Plancherel formula (in the repulsive and attractive regime). This important paper should be cited.
2- Typo: Page 6, figure 3: morion --> motion
3- Discuss possible extension to interacting case (sea Weakness)