Calogero-Moser System Research Articles

Generalized Calogero-Moser system and supergroup gauge origami

We study the integrability and the Bethe/Gauge correspondence of the Generalized Calogero-Moser system proposed by Berntson, Langmann and Lenells [1] which we call the elliptic quadruple Calogero-Moser system (eqCM). We write down the Dunkl operators which give commuting Hamiltonians of the quantum integrable system. We identify the gauge theory in correspondence is a supergroup version of the gauge origami, from which we construct the transfer matrix of the eqCM system.

Baxter Q-operator for the hyperbolic Calogero–Moser system *

We introduce a Q-operator Qz for the hyperbolic Calogero–Moser system as a one-parameter family of explicit integral operators. We establish the standard properties of a Q-operator, i.e. invariance of Hamiltonians, commutativity for different parameter values and that its eigenvalues satisfy an explicitly given first order ordinary difference equation in the parameter z.

Integrable scattering theory with higher derivative Hamiltonians

We discuss how a standard scattering theory a of multi-particle theory generalises to systems based on Hamiltonians that involve higher-order derivatives in their quantum mechanical formulation. As concrete examples, we consider Hamiltonian systems built from higher-order charges of Calogero and Calogero-Moser systems. Exploiting the integrability of these systems, we compute the classical phase shifts and briefly comment on the quantum versions of these types of theories.

Quantum generalized Calogero–Moser systems from free Hamiltonian reduction

The one-dimensional system of particles with a 1/x2 repulsive potential is known as the Calogero–Moser system. Its classical version can be generalized by substituting the coupling constants with additional degrees of freedom, which span the so(N) or su(N) algebra with respect to Poisson brackets. We present the quantum version of this generalized model. As the classical generalization is obtained by a symplectic reduction of a free system, we present a method of obtaining a quantum system along similar lines. The reduction of a free quantum system results in a Hamiltonian, which preserves the differences in dynamics of the classical system depending on the underlying, orthogonal or unitary, symmetry group. The orthogonal system is known to be less repulsive than the unitary one, and the reduced free quantum Hamiltonian manifests this trait through an additional attractive term ∑i<j−ħ2(xi−xj)2, which is absent when one performs the straightforward canonical Dirac quantization of the considered system. We present a detailed and rigorous derivation of the generalized quantum Calogero–Moser Hamiltonian, we find the spectra and wavefunctions for the number of particles N=2,3, and we diagonalize the Hamiltonian partially for a general value of N.

A dynamical interpretation of sequential decay in reactive scattering

The topic of this article is a dynamical explanation of the sequential decay in rearrangement scattering. The essential observation is the behaviour of trajectories close to the basin boundary of the breakup channel. As a most simplistic example of demonstration, we use a version of the perturbed three particle Calogero–Moser system in a 1-dimensional position space.

Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Bäcklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin–Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

Consistency of the Bäcklund Transformation for the Spin Calogero–Moser System

We prove the consistency of the Bäcklund transformation (BT) for the spin Calogero–Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ordinary differential equations; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency. We show that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.

Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics

We present a class of periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation, which was recently introduced by the authors together with Langmann as a classical, continuum limit of an Inozemtsev-type spin chain. These exact analytic solutions are constructed via a spin-pole ansatz written in terms of certain elliptic functions. The dynamical parameters in our solutions solve an elliptic spin Calogero-Moser (CM) system subject to certain constraints. In the course of our construction, we establish a novel Bäcklund transformation for this constrained elliptic spin CM system.

Stationary Flows Revisited

In this paper we revisit the subject of stationary flows of Lax hierarchies of a coupled KdV class. We explain the main ideas in the standard KdV case and then consider the dispersive water waves (DWW) case, with respectively 2 and 3 Hamiltonian representations. Each Hamiltonian representation gives us a different form of stationary flow. Comparing these, we construct Poisson maps, which, being non-canonical, give rise to bi-Hamiltonian representations of the stationary flows. An alternative approach is to use the Miura maps, which we do in the case of the DWW hierarchy, which has two ''modifications''. This structure gives us 3 sequences of Poisson related stationary flows. We use the Poisson maps to build a tri-Hamiltonian representation of each of the three stationary hierarchies. One of the Hamiltonian representations allows a multi-component squared eigenfunction expansion, which gives N degrees of freedom Hamiltonians, with first integrals. A Lax representation for each of the stationary flows is derived from the coupled KdV matrices. In the case of 3 degrees of freedom, we give a generalisation of our Lax matrices and Hamiltonian functions, which allows a connection with the rational Calogero-Moser (CM) system. This gives a coupling of the CM system with other potentials, along with a Lax representation. We present the particular case of coupling one of the integrable Hénon-Heiles systems to CM.

Lagrangian multiforms on Lie groups and non-commuting flows

We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational description of integrable systems in the sense of multidimensional consistency. In the context of non-commuting flows, the manifold of independent variables, often called multi-time, is a Lie group whose bracket structure corresponds to the commutation relations between the vector fields generating the flows. Natural examples are provided by superintegrable systems for the case of Lagrangian 1-form structures, and integrable hierarchies on loop groups in the case of Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the rational Calogero-Moser system, and a generalisation of the Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this endeavour as a first step towards a purely variational approach to Lie group actions on manifolds.

Noncommutative Bispectral Algebras and Their Presentations

We prove a general result on presentations of finitely generated algebras and apply it to obtain nice presentations for some noncommutative algebras arising in the matrix bispectral problem. By “nice presentation”, we mean a presentation that has as few as possible defining relations. This, in turn, has potential applications in computer algebra implementations and examples. Our results can be divided into three parts. In the first two, we consider bispectral algebras with the eigenvalue in the physical equation to be scalar-valued for 2×2 and 3×3 matrix-valued eigenfunctions. In the third part, we assume the eigenvalue in the physical equation to be matrix-valued and draw an important connection with Spin Calogero–Moser systems. In all cases, we show that these algebras are finitely presented. As a byproduct, we answer positively a conjecture of F. A. Grünbaum about these algebras.

Matrix and vectorial generalized Calogero–Moser models

Many-body one-dimensional systems with 1/x2 interactions are known as Calogero–Moser systems. The so called ordinary system with a common coupling constant for all interacting pairs can be generalized into models with couplings which evolve as additional degrees of freedom. This can be done via unitary reduction of a linear matrix model or, alternatively, by assigning a vectorial degree of freedom to each particle. We briefly review the matrix and vector formulations in the context of Hamiltonian mechanics. Various models of this type were investigated separately in several previous studies. In this paper we focus on relations among them. In particular we prove that the seemingly two-particle variables governing the repulsion of each pair are in fact functions of one-particle variables. Identification of the ordinary Calogero–Moser system as an isolated stationary point in the space of generalized models is a starting point to classifying the generalized models according to the rank of the matrix of couplings or, equivalently, the effective number of independent non-spacial degrees of freedom. Finally, we present two new examples of generalized Calogero–Moser models. The first one combines the matrix and vector degrees of freedom and results in additional 1/x interaction terms. The second one stems from a Hamilton function which mimics the Hamiltonian of a quantum model with spin exchange terms and leads to matrix anticommutator equations of motion.

Algebra of Dunkl Laplace–Runge–Lenz vector

We introduce the Dunkl version of the Laplace–Runge–Lenz vector associated with a finite Coxeter group W acting geometrically in mathbb R^N and with a multiplicity function g. This vector generalizes the usual Laplace–Runge–Lenz vector and its components commute with the Dunkl–Coulomb Hamiltonian given as the Dunkl Laplacian with an additional Coulomb potential gamma /r. We study the resulting symmetry algebra R_{g, gamma }(W) and show that it has the Poincaré–Birkhoff–Witt property. In the absence of a Coulomb potential, this symmetry algebra R_{g,0}(W) is a subalgebra of the rational Cherednik algebra H_g(W). We show that a central quotient of the algebra R_{g, gamma }(W) is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra H_g^{so(N+1)}(W). This gives an interpretation of the algebra H_g^{so(N+1)}(W) as the hidden symmetry algebra of the Dunkl–Coulomb problem in mathbb {R}^N. By specialising R_{g, gamma }(W) to g=0, we recover a quotient of the universal enveloping algebra U(so(N+1)) as the hidden symmetry algebra of the Coulomb problem in {mathbb R}^N. We also apply the Dunkl Laplace–Runge–Lenz vector to establish the maximal superintegrability of the generalised Calogero–Moser systems.

Integrable and superintegrable extensions of the rational Calogero–Moser model in three dimensions

We consider a class of Hamiltonian systems in 3 degrees of freedom, with a particular type of quadratic integral and which includes the rational Calogero–Moser system as a particular case. For the general class, we introduce separation coordinates to find the general separable (and therefore Liouville integrable) system, with two quadratic integrals. This gives a coupling of the Calogero–Moser system with a large class of potentials, generalising the series of potentials which are separable in parabolic coordinates. Particular cases are superintegrable, including Kepler and a resonant oscillator. The initial calculations of the paper are concerned with the flat (Cartesian type) kinetic energy, but in section 5, we introduce a conformal factor φ to H and extend the two quadratic integrals to this case. All the previous results are generalised to this case. We then introduce some two and three dimensional symmetry algebras of the Kinetic energy (Killing vectors), which restrict the conformal factor. This enables us to reduce our systems from 3 to 2 degrees of freedom, giving rise to many interesting systems, including both Kepler type and Hénon–Heiles type potentials on a Darboux–Koenigs D 2 background.

The non-chiral intermediate Heisenberg ferromagnet equation

We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter δ > 0, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit δ → ∞ it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of an Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.

N-point spherical functions and asymptotic boundary KZB equations

Let G be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on G in which the role of the quasi-simple admissible G-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of G. We show that they produce eigenstates for spin versions of quantum hyperbolic Calogero–Moser systems. In the second part of the paper we define and study special subclasses of global and formal elementary spherical functions, which we call global and formal N-point spherical functions. Formal N-point spherical functions arise as limits of correlation functions for boundary Wess–Zumino–Witten conformal field theory on the cylinder when the position variables tend to infinity. We construct global N-point spherical functions in terms of compositions of equivariant differential intertwiners associated with principal series representations, and express them in terms of Eisenstein integrals. We show that the eigenstates of the quantum spin Calogero–Moser system associated to N-point spherical functions are also common eigenfunctions of a commuting family of first-order differential operators, which we call asymptotic boundary Knizhnik–Zamolodchikov–Bernard operators. These operators are explicitly given in terms of folded classical dynamical r-matrices and associated dynamical k-matrices.

Novikov-Veselov Symmetries of the Two-Dimensional O(N) Sigma Model

We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries of two-dimensional O(N) sigma model. In the first part of the paper we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case of irreducible Fermi curves is complete. In the second part of the paper we generalize our construction to the case of reducible Fermi curves and show that it gives the conformal harmonic maps to even-dimensional spheres. Remarkably, the solutions are parameterized by spectral curves of turning points of the elliptic Calogero-Moser system.

Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Generalizations of parabolic Higgs bundles, real structures, and integrability

We introduce a notion of quasi-antisymmetric Higgs $G$-bundles over curves with marked points. They are endowed with additional structures, which replace the parabolic structures at marked points in the parabolic Higgs bundles. The latter means that the coadjoint orbits are attached to the marked points. The moduli spaces of parabolic Higgs bundles are the phase spaces of complex completely integrable systems. In our case the coadjoint orbits are replaced by the cotangent bundles over some special symmetric spaces in such a way that the moduli space of the modified Higgs bundles are still phase spaces of complex completely integrable systems. We show that the moduli space of the parabolic Higgs bundles is the symplectic quotient of the moduli space of the quasi-antisymmetric Higgs bundle with respect to the action of product of Cartan subgroups. Also, by changing the symmetric spaces we introduce quasi-compact and quasi-normal Higgs bundles. Then the fixed point sets of real involutions acting on their moduli spaces are the phase spaces of real completely integrable systems. Several examples are given including integrable extensions of the ${\rm SL}(2)$ Euler-Arnold top, two-body elliptic Calogero-Moser system and the rational ${\rm SL}(2)$ Gaudin system together with its real reductions.

Superintegrability of Calogero–Moser systems associated with the cyclic quiver

We study complex integrable systems on quiver varieties associated with the cyclic Noquiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero–Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type A n−1 and B n , as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.