9,553 research outputs found
Integrable Time-Discretisation of the Ruijsenaars-Schneider Model
An exactly integrable symplectic correspondence is derived which in a
continuum limit leads to the equations of motion of the relativistic
generalization of the Calogero-Moser system, that was introduced for the first
time by Ruijsenaars and Schneider. For the discrete-time model the equations of
motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2
Heisenberg magnet. We present a Lax pair, the symplectic structure and prove
the involutivity of the invariants. Exact solutions are investigated in the
rational and hyperbolic (trigonometric) limits of the system that is given in
terms of elliptic functions. These solutions are connected with discrete
soliton equations. The results obtained allow us to consider the Bethe Ansatz
equations as ones giving an integrable symplectic correspondence mixing the
parameters of the quantum integrable system and the parameters of the
corresponding Bethe wavefunction.Comment: 27 pages, latex, equations.st
Generalized Lame operators
We introduce a class of multidimensional Schr\"odinger operators with
elliptic potential which generalize the classical Lam\'e operator to higher
dimensions. One natural example is the Calogero--Moser operator, others are
related to the root systems and their deformations. We conjecture that these
operators are algebraically integrable, which is a proper generalization of the
finite-gap property of the Lam\'e operator. Using earlier results of Braverman,
Etingof and Gaitsgory, we prove this under additional assumption of the usual,
Liouville integrability. In particular, this proves the Chalykh--Veselov
conjecture for the elliptic Calogero--Moser problem for all root systems. We
also establish algebraic integrability in all known two-dimensional cases. A
general procedure for calculating the Bloch eigenfunctions is explained. It is
worked out in detail for two specific examples: one is related to B_2 case,
another one is a certain deformation of the A_2 case. In these two cases we
also obtain similar results for the discrete versions of these problems,
related to the difference operators of Macdonald--Ruijsenaars type.Comment: 38 pages, latex; in the new version a reference was adde
A remark on the Hankel determinant formula for solutions of the Toda equation
We consider the Hankel determinant formula of the functions of the
Toda equation. We present a relationship between the determinant formula and
the auxiliary linear problem, which is characterized by a compact formula for
the functions in the framework of the KP theory. Similar phenomena that
have been observed for the Painlev\'e II and IV equations are recovered. The
case of finite lattice is also discussed.Comment: 14 pages, IOP styl
Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems
Classically, a single weight on an interval of the real line leads to
moments, orthogonal polynomials and tridiagonal matrices. Appropriately
deforming this weight with times t=(t_1,t_2,...), leads to the standard Toda
lattice and tau-functions, expressed as Hermitian matrix integrals.
This paper is concerned with a sequence of t-perturbed weights, rather than
one single weight. This sequence leads to moments, polynomials and a (fuller)
matrix evolving according to the discrete KP-hierarchy. The associated
tau-functions have integral, as well as vertex operator representations.
Among the examples considered, we mention: nested Calogero-Moser systems,
concatenated solitons and m-periodic sequences of weights. The latter lead to
2m+1-band matrices and generalized orthogonal polynomials, also arising in the
context of a Riemann-Hilbert problem.
We show the Riemann-Hilbert factorization is tantamount to the factorization
of the moment matrix into the product of a lower- times upper-triangular
matrix.Comment: 40 page
Quantum Calogero-Moser Models: Integrability for all Root Systems
The issues related to the integrability of quantum Calogero-Moser models
based on any root systems are addressed. For the models with degenerate
potentials, i.e. the rational with/without the harmonic confining force, the
hyperbolic and the trigonometric, we demonstrate the following for all the root
systems: (i) Construction of a complete set of quantum conserved quantities in
terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal
R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the
Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of
the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack
polynomials are defined for all root systems as unique eigenfunctions of the
Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v)
Algebraic construction of all excited states in terms of creation operators.
These are mainly generalisations of the results known for the models based on
the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure
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