Abstract
Abstract In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic relativistic Calogero–Moser type. We focused on the 1st steps of the scheme in Part I and on the cases $N=2$ and $N=3$ in Part II. In this paper, we determine the dominant asymptotics of a similarity-transformed function $\textrm{E}_N(b;x,y)$ for $y_j-y_{j+1}\to \infty $, $j=1,\ldots , N-1$ and thereby confirm the long-standing conjecture that the particles in the hyperbolic relativistic Calogero–Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers $N>3$.
Highlights
1 Introduction This paper is the 3rd part in a series of papers dedicated to the explicit diagonalization and Hilbert space transform theory for the integrable N-particle systems of hyperbolic relativistic Calogero–Moser type
The asymptotic behavior (31) confirms a long-standing conjecture. It says that the particles in the relativistic Calogero–Moser systems of hyperbolic type exhibit soliton scattering, cf
Factorized asymptotics for the hyperbolic case with arbitrary positive coupling was first proved by Opdam [8], working within the arbitrary root system context developed by him and Heckman, a summary of which can be found in [7]
Summary
This paper is the 3rd part in a series of papers dedicated to the explicit diagonalization and Hilbert space transform theory for the integrable N-particle systems of hyperbolic relativistic Calogero–Moser type. We shall make use of previous results in this series of papers without further ado, referring back to sections and equations in [4] and [5] by using the prefix I and II, respectively. The asymptotic behavior (31) confirms a long-standing conjecture In physical parlance, it says that the particles in the relativistic Calogero–Moser systems of hyperbolic type exhibit soliton scattering (conservation of momenta and factorization of the S-matrix), cf I Section 7. Factorized asymptotics for the hyperbolic case with arbitrary positive coupling was first proved by Opdam [8], working within the arbitrary root system context developed by him and Heckman, a summary of which can be found in [7]. As in our previous work, a key point is rather to use their recursive structure
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