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TMF, 2012, Volume 172, Number 2, Pages 198–223 (Mi tmf6942)  

This article is cited in 5 scientific papers (total in 5 papers)

On a technique to identify solvable discrete-time many-body problems

F. Calogeroab

a Physics Department, University of Rome "La Sapienza", Romе, Italy
b Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Romе, Italy

Abstract: The starting point is an $N{\times}N$ matrix, $U\equiv U(\ell)$, evolving in the discrete-time independent variable $\ell=0,1,2,…$ 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.

Keywords: integrable discrete-time dynamical system, solvable discrete-time dynamical system, integrable discrete-time many-body problem, solvable discrete-time many-body problem, isochronous discrete-time evolution

DOI: https://doi.org/10.4213/tmf6942

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English version:
Theoretical and Mathematical Physics, 2012, 172:2, 1052–1072

Bibliographic databases:


Citation: F. Calogero, “On a technique to identify solvable discrete-time many-body problems”, TMF, 172:2 (2012), 198–223; Theoret. and Math. Phys., 172:2 (2012), 1052–1072

Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf6942
  • http://mi.mathnet.ru/eng/tmf/v172/i2/p198

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. F. Calogero, F. Leyvraz, “New solvable discrete-time many-body problem featuring several arbitrary parameters”, J. Math. Phys., 53:8 (2012), 082702, 19 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. F. Calogero, F. Leyvraz, “New solvable discrete-time many-body problem featuring several arbitrary parameters. II”, J. Math. Phys., 54:10 (2013), 102702  crossref  mathscinet  zmath  adsnasa  isi
    3. M. Bruschi, F. Calogero, F. Leyvraz, “A large class of solvable discrete-time many-body problems”, J. Math. Phys., 55:8 (2014), 082703  crossref  mathscinet  zmath  adsnasa  isi
    4. F. Calogero, F. Leyvraz, “A nonautonomous yet solvable discrete-time $n$-body problem”, J. Phys. A-Math. Theor., 47:10 (2014), 105203  crossref  mathscinet  zmath  adsnasa  isi
    5. O. Bihun, F. Calogero, “Generations of solvable discrete-time dynamical systems”, J. Math. Phys., 58:5 (2017), 052701  crossref  mathscinet  zmath  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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