93 research outputs found
Bethe ansatz for the Ruijsenaars model of BC1- type
We consider one-dimensional elliptic Ruijsenaars model of type BC1. We show that when all coupling constants are integers, it has meromorphic eigenfunctions expressed by a variant of Bethe ansatz. This result generalizes the Bethe ansatz formulas known in the A1-case
On algebraic integrability of the deformed elliptic Calogero--Moser problem
Algebraic integrability of the elliptic Calogero--Moser quantum problem
related to the deformed root systems \pbf{A_{2}(2)} is proved. Explicit
formulae for integrals are found
A remark on rational isochronous potentials
We consider the rational potentials of the one-dimensional mechanical
systems, which have a family of periodic solutions with the same period
(isochronous potentials). We prove that up to a shift and adding a constant all
such potentials have the form or Comment: 5 pages, contribution to a special issue of JNMP dedicated to F.
Calogero, slightly revised versio
Quantum Lax Pairs via Dunkl and Cherednik Operators
We establish a direct link between Dunkl operators and quantum Lax matrices L for the Calogero–Moser systems associated to an arbitrary Weyl group W (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix A so that L,A form a quantum Lax pair. Moreover, such an A can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of W, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D’Hoker–Phong and Bordner–Corrigan–Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic BCn case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem
On generalisations of Calogero-Moser-Sutherland quantum problem and WDVV equations
It is proved that if the Schr\"odinger equation of
Calogero-Moser-Sutherland type with
has a solution of the product form then the function satisfies the
generalised WDVV equations.Comment: 10 page
Multidimensional Baker-Akhiezer functions and Huygens' Principle
A notion of rational Baker-Akhiezer (BA) function related to a configuration
of hyperplanes in C^n is introduced. It is proved that BA function exists only
for very special configurations (locus configurations), which satisfy certain
overdetermined algebraic system. The BA functions satisfy some algebraically
integrable Schrodinger equations, so any locus configuration determines such an
equation. Some results towards the classification of all locus configurations
are presented. This theory is applied to the famous Hadamard's problem of
description of all hyperbolic equations satisfying Huygens' Principle. We show
that in a certain class all such equations are related to locus configurations
and the corresponding fundamental solutions can be constructed explicitly from
the BA functions.Comment: 35 pages, LATEX, 2 figures included in graphicx. Submitted to
Comm.Math.Phys. (Dec. 1998
Bethe Ansatz for the Ruijsenaars Model of BC₁-Type
We consider one-dimensional elliptic Ruijsenaars model of type BC₁. It is given by a three-term difference Schrödinger operator L containing 8 coupling constants. We show that when all coupling constants are integers, L has meromorphic eigenfunctions expressed by a variant of Bethe ansatz. This result generalizes the Bethe ansatz formulas known in the A₁-case
Quantum integrability of the deformed elliptic Calogero-Moser problem
The integrability of the deformed quantum elliptic Calogero-Moser problem
introduced by Chalykh, Feigin and Veselov is proven. Explicit recursive
formulae for the integrals are found. For integer values of the parameter this
implies the algebraic integrability of the systems.Comment: 23 page
Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models
We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with m vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case m = 1 corresponds to the tadpole quiver and the Ruijsenaars–Schneider system and its variants, while for m > 1 we obtain new integrable systems that generalise the Ruijsenaars–Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs (Braverman et al. [32,34,35] and Kodera and Nakajima [36]), as well as in the context of the Macdonald theory (Chalykh and Etingof, 2013)
N=4 Mechanics, WDVV Equations and Polytopes
N=4 superconformal n-particle quantum mechanics on the real line is governed
by two prepotentials, U and F, which obey a system of partial nonlinear
differential equations generalizing the Witten-Dijkgraaf-Verlinde-Verlinde
(WDVV) equation for F. The solutions are encoded by the finite Coxeter systems
and certain deformations thereof, which can be encoded by particular polytopes.
We provide A_n and B_3 examples in some detail. Turning on the prepotential U
in a given F background is very constrained for more than three particles and
nonzero central charge. The standard ansatz for U is shown to fail for all
finite Coxeter systems. Three-particle models are more flexible and based on
the dihedral root systems.Comment: Talk at ISQS-17 in Prague, 19-21 June 2008, and at Group-27 in
Yerevan, 13-19 August 2008; v2: B_3 examples correcte
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