New solvable discrete-time many-body problem featuring several arbitrary parameters (Q2872239)

This paper is devoted to present a family of solvable discrete-time many-body problems, depending on several parameters.NEWLINENEWLINEThe construction of such a family uses a generalization to systems with discrete time of the Olshanestky-Perelomov technique. It is based on considering a solvable first-order system of two discrete-time evolution equations involving two \(N\times N\) matrices. The solvable discrete-time many-body problem is given by the evolution of the eigenvalues of those matrices.NEWLINENEWLINEThe family of solvable discrete-time problems starts from the five-parameter system for \(N\times N\) matrices \(U(\ell)\) and \(V(\ell)\) NEWLINE\[NEWLINE \begin{aligned} U \tilde U & = \alpha \tilde U + \beta U + \gamma V + \delta V \tilde U + \eta UV, \\ \tilde V & = V, \end{aligned} NEWLINE\]NEWLINE where \(\tilde W(\ell) = W(\ell+1)\) and \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), and \(\eta\) are constant. After rescaling, only three parameters are free. The fact that this system is solvable allows the authors to identify a three-parameter family of discrete-time many-body problems, whose deduction is carried out.NEWLINENEWLINEThe authors also provide a partial study of the long time behavior of the solutions of the solvable model. They identify a particular choice of the parameters which lead to asymptotically isochronous solutions, that is, NEWLINE\[NEWLINE \lim_{\ell\to \infty } z (\ell + L) - z(\ell) = 0, NEWLINE\]NEWLINE where \(z\) are the coordinates of the many-body problem induced by the matrix system above. Other choices of the parameters lead to an almost periodic behavior both in the past and in the future.NEWLINENEWLINEFinally, the authors present a possible generalization of the system above by considering the parameters depending on the discrete time.