300 research outputs found
Continuum limit of the Volterra model, separation of variables and non standard realizations of the Virasoro Poisson bracket
The classical Volterra model, equipped with the Faddeev-Takhtadjan Poisson
bracket provides a lattice version of the Virasoro algebra. The Volterra model
being integrable, we can express the dynamical variables in terms of the so
called separated variables. Taking the continuum limit of these formulae, we
obtain the Virasoro generators written as determinants of infinite matrices,
the elements of which are constructed with a set of points lying on an infinite
genus Riemann surface. The coordinates of these points are separated variables
for an infinite set of Poisson commuting quantities including . The
scaling limit of the eigenvector can also be calculated explicitly, so that the
associated Schroedinger equation is in fact exactly solvable.Comment: Latex, 43 pages Synchronized with the to be published versio
The symplectic structure of rational Lax pair systems
We consider dynamical systems associated to Lax pairs depending rationnally
on a spectral parameter. We show that we can express the symplectic form in
terms of algebro--geometric data provided that the symplectic structure on L is
of Kirillov type. In particular, in this case the dynamical system is
integrable.Comment: 8 pages, no figure, Late
Dressing Symmetries
We study Lie-Poisson actions on symplectic manifolds. We show that they are
generated by non-Abelian Hamiltonians. We apply this result to the group of
dressing transformations in soliton theories; we find that the non-Abelian
Hamiltonian is just the monodromy matrix. This provides a new proof of their
Lie-Poisson property. We show that the dressing transformations are the
classical precursors of the non-local and quantum group symmetries of these
theories. We treat in detail the examples of the Toda field theories and the
Heisenberg model.Comment: (29 pages
The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems
We quantize the spin Calogero-Moser model in the -matrix formalism. The
quantum -matrix of the model is dynamical. This -matrix has already
appeared in Gervais-Neveu's quantization of Toda field theory and in Felder's
quantization of the Knizhnik-Zamolodchikov-Bernard equation.Comment: Comments and References adde
A semiclassical study of the Jaynes-Cummings model
We consider the Jaynes-Cummings model of a single quantum spin coupled to
a harmonic oscillator in a parameter regime where the underlying classical
dynamics exhibits an unstable equilibrium point. This state of the model is
relevant to the physics of cold atom systems, in non-equilibrium situations
obtained by fast sweeping through a Feshbach resonance. We show that in this
integrable system with two degrees of freedom, for any initial condition close
to the unstable point, the classical dynamics is controlled by a singularity of
the focus-focus type. In particular, it displays the expected monodromy, which
forbids the existence of global action-angle coordinates. Explicit calculations
of the joint spectrum of conserved quantities reveal the monodromy at the
quantum level, as a dislocation in the lattice of eigenvalues. We perform a
detailed semi-classical analysis of the associated eigenstates. Whereas most of
the levels are well described by the usual Bohr-Sommerfeld quantization rules,
properly adapted to polar coordinates, we show how these rules are modified in
the vicinity of the critical level. The spectral decomposition of the
classically unstable state is computed, and is found to be dominated by the
critical WKB states. This provides a useful tool to analyze the quantum
dynamics starting from this particular state, which exhibits an aperiodic
sequence of solitonic pulses with a rather well defined characteristic
frequency.Comment: pdfLaTeX, 51 pages, 19 figures, references added and improved figure
captions. To appear in J. Stat. Mec
A Quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations
We consider the universal solution of the Gervais-Neveu-Felder equation in
the case. We show that it has a quasi-Hopf algebra
interpretation. We also recall its relation to quantum 3-j and 6-j symbols.
Finally, we use this solution to build a q-deformation of the trigonometric
Lam\'e equation.Comment: 9 pages, 4 figure
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