1,213 research outputs found
On frequencies of small oscillations of some dynamical systems associated with root systems
In the paper by F. Calogero and author [Commun. Math. Phys. 59 (1978)
109-116] the formula for frequencies of small oscillations of the Sutherland
system ( case) was found. In present note the generalization of this
formula for the case of arbitrary root system is given.Comment: arxiv version is already officia
The matrix Kadomtsev--Petviashvili equation as a source of integrable nonlinear equations
A new integrable class of Davey--Stewartson type systems of nonlinear partial
differential equations (NPDEs) in 2+1 dimensions is derived from the matrix
Kadomtsev--Petviashvili equation by means of an asymptotically exact nonlinear
reduction method based on Fourier expansion and spatio-temporal rescaling. The
integrability by the inverse scattering method is explicitly demonstrated, by
applying the reduction technique also to the Lax pair of the starting matrix
equation and thereby obtaining the Lax pair for the new class of systems of
equations. The characteristics of the reduction method suggest that the new
systems are likely to be of applicative relevance. A reduction to a system of
two interacting complex fields is briefly described.Comment: arxiv version is already officia
Goldfishing by gauge theory
A new solvable many-body problem of goldfish type is identified and used to
revisit the connection among two different approaches to solvable dynamical
systems. An isochronous variant of this model is identified and investigated.
Alternative versions of these models are presented. The behavior of the
alternative isochronous model near its equilibrium configurations is
investigated, and a remarkable Diophantine result, as well as related
Diophantine conjectures, are thereby obtained.Comment: 22 page
Knizhnik-Zamolodchikov equations and the Calogero-Sutherland-Moser integrable models with exchange terms
It is shown that from some solutions of generalized Knizhnik-Zamolodchikov
equations one can construct eigenfunctions of the Calogero-Sutherland-Moser
Hamiltonians with exchange terms, which are characterized by any given
permutational symmetry under particle exchange. This generalizes some results
previously derived by Matsuo and Cherednik for the ordinary
Calogero-Sutherland-Moser Hamiltonians.Comment: 13 pages, LaTeX, no figures, to be published in J. Phys.
Generalization of a result of Matsuo and Cherednik to the Calogero-Sutherland- Moser integrable models with exchange terms
A few years ago, Matsuo and Cherednik proved that from some solutions of the
Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field
theory, one can obtain wave functions for the Calogero integrable system. In
the present communication, it is shown that from some solutions of generalized
KZ equations, one can construct wave functions, characterized by any given
permutational symmetry, for some Calogero-Sutherland-Moser integrable models
with exchange terms. Such models include the spin generalizations of the
original Calogero and Sutherland ones, as well as that with -function
interaction.Comment: Latex, 7 pages, Communication at the 4th Colloquium "Quantum Groups
and Integrable Systems", Prague (June 1995
Upper and lower limits on the number of bound states in a central potential
In a recent paper new upper and lower limits were given, in the context of
the Schr\"{o}dinger or Klein-Gordon equations, for the number of S-wave
bound states possessed by a monotonically nondecreasing central potential
vanishing at infinity. In this paper these results are extended to the number
of bound states for the -th partial wave, and results are also
obtained for potentials that are not monotonic and even somewhere positive. New
results are also obtained for the case treated previously, including the
remarkably neat \textit{lower} limit with (valid in the Schr\"{o}dinger case, for a class of potentials
that includes the monotonically nondecreasing ones), entailing the following
\textit{lower} limit for the total number of bound states possessed by a
monotonically nondecreasing central potential vanishing at infinity: N\geq
\{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2 (here the double braces denote of
course the integer part).Comment: 44 pages, 5 figure
Poisson Structures for Aristotelian Model of Three Body Motion
We present explicitly Poisson structures, for both time-dependent and
time-independent Hamiltonians, of a dynamical system with three degrees of
freedom introduced and studied by Calogero et al [2005]. For the
time-independent case, new constant of motion includes all parameters of the
system. This extends the result of Calogero et al [2009] for semi-symmetrical
motion. We also discuss the case of three bodies two of which are not
interacting with each other but are coupled with the interaction of third one
Crossover from Fermi Liquid to Non-Fermi Liquid Behavior in a Solvable One-Dimensional Model
We consider a quantum moany-body problem in one-dimension described by a
Jastrow type, characterized by an exponent and a parameter .
We show that with increasing , the Fermi Liquid state (
crosses over to non-Fermi liquid states, characterized by effective
"temperature".Comment: 8pp. late
Testing Hall-Post Inequalities With Exactly Solvable N-Body Problems
The Hall--Post inequalities provide lower bounds on -body energies in
terms of -body energies with . They are rewritten and generalized to
be tested with exactly-solvable models of Calogero-Sutherland type in one and
higher dimensions. The bound for spinless fermions in one dimension is
better saturated at large coupling than for noninteracting fermions in an
oscillatorComment: 7 pages, Latex2e, 2 .eps figure
Exchange Operator Formalism for Integrable Systems of Particles
We formulate one dimensional many-body integrable systems in terms of a new
set of phase space variables involving exchange operators. The hamiltonian in
these variables assumes a decoupled form. This greatly simplifies the
derivation of the conserved charges and the proof of their commutativity at the
quantum level.Comment: 8 page
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