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Two novel classes of solvable many-body problems of goldfish type with constraints

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dc.contributor.authorCalogero, Francesco C.
dc.contributor.authorGómez-Ullate Otaiza, David
dc.date.accessioned2023-06-20T10:55:10Z
dc.date.available2023-06-20T10:55:10Z
dc.date.issued2007-05-18
dc.description© IOP Publishing Ltd. The results reported in this paper where obtained during a visit in October 2006 of one of us (DGU) to the Department of Physics of the University of Rome “La Sapienza”, performed in the context of the Collaboration Agreement among the University of Rome “La Sapienza” and the Universidad Complutense of Madrid. The research of DGU is supported in part by the Ramón y Cajal program of the Ministerio de Ciencia y Tecnología and by the DGI under grants FIS2005-00752 and MTM2006-00478. It is a pleasure to acknowledge illuminating discussions with Robert Milson.
dc.description.abstractTwo novel classes of many-body models with nonlinear interactions 'of goldfish type' are introduced. They are solvable provided the initial data satisfy a single constraint (in one case; in the other, two constraints), i.e., for such initial data the solution of their initial-value problem can be achieved via algebraic operations, such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials. Entirely isochronous versions of some of these models are also exhibited, i.e., versions of these models whose nonsingular solutions are all completely periodic with the same period.
dc.description.departmentDepto. de Física Teórica
dc.description.facultyFac. de Ciencias Físicas
dc.description.refereedTRUE
dc.description.sponsorshipRamón y Cajal program of the Ministerio de Ciencia y Tecnología
dc.description.sponsorshipDGI
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/30843
dc.identifier.doi10.1088/1751-8113/40/20/007
dc.identifier.issn1751-8113
dc.identifier.officialurlhttp://dx.doi.org/10.1088/1751-8113/40/20/007
dc.identifier.relatedurlhttp://iopscience.iop.org
dc.identifier.relatedurlhttp://arxiv.org/abs/nlin/0701046
dc.identifier.urihttps://hdl.handle.net/20.500.14352/51450
dc.issue.number20
dc.journal.titleJournal of physics A: Mathematical and theoretical
dc.language.isoeng
dc.page.final5353
dc.page.initial5335
dc.publisherIOP Publishing Ltd
dc.relation.projectIDFIS2005-00752
dc.relation.projectIDMTM2006-00478
dc.rights.accessRightsopen access
dc.subject.cdu51-73
dc.subject.keywordQuasi-exact solvability
dc.subject.keywordSubspaces
dc.subject.ucmFísica-Modelos matemáticos
dc.subject.ucmFísica matemática
dc.titleTwo novel classes of solvable many-body problems of goldfish type with constraints
dc.typejournal article
dc.volume.number40
dcterms.references[1] F. Calogero, “Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations, and Related “Solvable” Many Body Problems”, Nuovo Cimento 43B, 177-241 (1978). [2] F. Calogero, Classical many-body problems amenable to exact treatments, Lecture Notes in Physics Monograph m 66, Springer, 2001. [3] F. Calogero, The “neatest” many-body problem amenable to exact treatments (a “goldfish”?), Physica D 152-153, 78-84 (2001). [4] F. Calogero, Isochronous systems, 200-page monograph, to be published, 2007. [5] F. Calogero and D. G´omez-Ullate, “Additional classes of solvable manybody problems of goldfish type with constraints”, in preparation. [6] D. Gómez-Ullate, N. Kamran, R. Milson, “The Darboux transformation and algebraic deformations of shape-invariant potentials”, J. Phys. A 37, 1789–1804 (2004). [7] D. Gómez-Ullate, N. Kamran, R. Milson, “Quasi-exact solvability and the direct approach to invariant subspaces”, J. Phys. A 38, 2005—2019 (2005). [8] D. Gómez-Ullate, N. Kamran, R. Milson, “Supersymmetry and algebraic Darboux transformations”, J. Phys. A 37, 10065–10078 (2004). [9] D. Gómez-Ullate, N. Kamran, R. Milson, “Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces, Discrete and Cont. Din. Syst., in press, arXiV:nlin.SI/0604070. [10] D. Gómez-Ullate, N. Kamran, and R. Milson, Quasi-exact solvability beyond the SL(2) algebraization, Phys. Atom. Nuclei, in press arXiv:nlin.SI/0601053. [11] D. Gómez-Ullate, N. Kamran, and R. Milson, “A generalized Bochner problem and related orthogonal polynomials”, in preparation. [12] D. Gómez-Ullate and M. Sommacal, “Periods of the goldfish many-body problem”, J. Nonlinear Math. Phys. 12, Suppl. 1, 351-362 (2005). [13] E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin (1987), p. 193-195.
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