10,584 research outputs found
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
Stochastic B\"acklund transformations
How does one introduce randomness into a classical dynamical system in order
to produce something which is related to the `corresponding' quantum system? We
consider this question from a probabilistic point of view, in the context of
some integrable Hamiltonian systems
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
On a Trivial Family of Noncommutative Integrable Systems
We discuss trivial deformations of the canonical Poisson brackets associated
with the Toda lattices, relativistic Toda lattices, Henon-Heiles, rational
Calogero-Moser and Ruijsenaars-Schneider systems and apply one of these
deformations to construct a new trivial family of noncommutative integrable
systems
- …