Abstract:
The starting point is an $N{\times}N$ matrix, $U\equiv U(\ell)$, evolving in the discrete-time independent variable $\ell=0,1,2,\dots$ according to a solvable matrix evolution equation. One then focuses on the evolution of its $N$ eigenvalues $z_n(\ell)$. This evolution generally also involves $N(N{-}1)$ additional variables. In some cases via a compatible ansatz these additional variables can be expressed in terms of the $N$ variables $z_n(\ell)$. Thereby one obtains a system of discrete-time evolution equations involving only the $N$ dependent variables $z_n(\ell)$, which is often interpretable as a discrete-time many-body problem. Various peculiarities of this approach are investigated, including the possibility to manufacture nontrivial isochronous models (all solutions of which are periodic with the same period). These properties are illustrated via specific examples. In the process novel discrete-time many-body problems are exhibited.
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