It is shown that the $E_7$ trigonometric Olshanetsky-Perelomov Hamiltonian, when written in terms of the Fundamental Trigonometric Invariants (FTI), is in algebraic form, i.e., has polynomial coefficients, and preserves the infinite flag of polynomial spaces with the characteristic vector $\vec α= (1,2,2,2,3,3,4)$. Its flag coincides with one of the minimal characteristic vector for the $E_7$ rational model.
A multivariable generalisation of the Bessel polynomials is introduced and studied. In particular, we deduce their series expansion in Jack polynomials, a limit transition from multivariable Jacobi polynomials, a sequence of algebraically independent eigenoperators, Pieri type recurrence relations, and certain orthogonality properties. We also show that these multivariable Bessel polynomials provide a (finite) set of eigenfunctions of the hyperbolic Sutherland model with external Morse potential.
In the previous paper math-ph/0507015 we have studied the characters and Clebsch-Gordan series for the exceptional Lie algebra E7 by relating them to the quantum trigonometric Calogero-Sutherland Hamiltonian with coupling constant K=1. Now we extend that approach to the case of general K.
Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The invariants of the basis can be used as coordinates in any cell of the coroot space and lead to an exactly solvable model of Sutherland type. We apply this construction to the $F_4$ case
We provide a list of explicit eigenfunctions of the trigonometric Calogero-Sutherland Hamiltonian associated to the root system of the exceptional Lie algebra E8. The quantum numbers of these solutions correspond to the first and second order weights of the Lie algebra.
We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model for the Lie algebra E8 and coupling constant k=1 by using the fundamental irreducible characters of the algebra as dynamical independent variables. Then, we compute the second order characters of the algebra and some higher order characters.
We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system ($\vee$-system) and we determine all trigonometric $\vee$-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric $\vee$-system; this inverts a one-way implication observed by Veselov for the rational solutions.
Quantum Calogero-Sutherland model of $A_n$ type is completely integrable. Using this fact, we give an elementary construction of lowering an raising operators for the trigonometric case. This is similar, but more complicated (due to the fact that the energy spectrum is not equidistant) than the construction for the rational case.
We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra $D_4$ in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra. This parametrization allows us to solve for the energy eigenfunctions of the theory and to study properties of the system of orthogonal polynomials associated to them such as recurrence relations and generating functions.
The quantum trigonometric Calogero-Sutherland models related to Lie algebras admit a parametrization in which the dynamical variables are the characters of the fundamental representations of the algebra. We develop here this approach for the case of the exceptional Lie algebra E6.
We report on the the trigonometric spin Ruijsenaars--Sutherland hierarchy derived recently by Poisson reduction of a bi-Hamiltonian hierarchy associated with free geodesic motion on the Lie group U(n). In particular, we give a direct proof of a previously stated result about the form of the second Poisson bracket in terms of convenient variables.
We re-express the quantum Calogero-Sutherland model for the Lie algebra E7 and the particular value of the coupling constant K=1 by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan series required to perform the change of variables. We describe how the resulting quantum Hamiltonian operator can be used to compute more characters and Clebsch-Gordan series for this exceptional algebra.
Recently Feher and the author have constructed the action-angle dual of the trigonometric BC(n) Sutherland system via Hamiltonian reduction. In this paper a reduction-based calculation is carried out to verify canonical Poisson bracket relations on the phase space of this dual model. Hence the material serves complementary purposes whilst it can also be regarded as a suitable modification of the hyperbolic case previously sorted out by Pusztai.
It is conjectured that any trigonometric Olshanetsky-Perelomov Hamiltonian written in Fundamental Trigonometric Invariants (FTI) as coordinates takes an algebraic form and preserves an infinite flag of spaces of polynomials. It is shown that try-and-guess variables which led to the algebraic form of trigonometric Olshanetsky-Perelomov Hamiltonians related to root spaces of the classical $A_N, B_N, C_N, D_N, BC_N$ and exceptional $G_2, F_4$ Lie algebras are FTI. This conjecture is also confirmed for the trigonometric $E_6$ Olshanetsky-Perelomov Hamiltonian whose algebraic form is found with the…
Starting from the Hamiltonian formulation of supersymmetric Calogero models associated with the classical $A_n$, $B_n$, $C_n$ and $D_n$ series we construct the ${\cal N}{=}\,2$ and ${\cal N}{=}\,4$ supersymmetric extensions of the their hyperbolic/trigonometric Calogero-Sutherland cousins. The bosonic core of these models are the standard Calogero-Sutherland hyperbolic/trigonometric systems.
It is shown that the $E_8$ trigonometric Olshanetsky-Perelomov Hamiltonian, when written in terms of the Fundamental Trigonometric Invariants (FTI), is in algebraic form, i.e., has polynomial coefficients, and preserves two infinite flags of polynomial spaces marked by the Weyl (co)-vector and $E_8$ highest root (both in the basis of simple roots) as characteristic vectors. The explicit form of the Hamiltonian in new variables has been obtained both by direct calculation and by means of the orbit function technique. It is shown a triangularity of the Hamiltonian in the bases of orbit functions…
We present a new case of duality between integrable many-body systems, where two systems live on the action-angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one. Our investigation utilizes an idea that was exploited previously to provide group-theoretic interpretation for several dualities discovered originally by Ruijsenaars. In the group-theoretic framework one applies Hamiltonian reduction to two Abelian Poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as act…
The BC(n) Sutherland Hamiltonian with coupling constants parametrized by three arbitrary integers is derived by reductions of the Laplace operator of the group U(N). The reductions are obtained by applying the Laplace operator on spaces of certain vector valued functions equivariant under suitable symmetric subgroups of U(N)\times U(N). Three different reduction schemes are considered, the simplest one being the compact real form of the reduction of the Laplacian of GL(2n,C) to the complex BC(n) Sutherland Hamiltonian previously studied by Oblomkov.
Besides its usual interpretation as a system of $n$ indistinguishable particles moving on the circle, the trigonometric Sutherland system can be viewed alternatively as a system of distinguishable particles on the circle or on the line, and these 3 physically distinct systems are in duality with corresponding variants of the rational Ruijsenaars-Schneider system. We explain that the 3 duality relations, first obtained by Ruijsenaars in 1995, arise naturally from the Kazhdan-Kostant-Sternberg symplectic reductions of the cotangent bundles of the group U(n) and its covering groups $U(1) \times S…
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.
Explicit solutions of the classical Calogero (rational with/without harmonic confining potential) and Sutherland (trigonometric potential) systems is obtained by diagonalisation of certain matrices of simple time evolution. The method works for Calogero & Sutherland systems based on any root system. It generalises the well-known results by Olshanetsky and Perelomov for the A type root systems. Explicit solutions of the (rational and trigonometric) higher Hamiltonian flows of the integrable hierarchy can be readily obtained in a similar way for those based on the classical root systems.
Integrable many-body systems of Ruijsenaars--Schneider--van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson-Lie group $\mathrm{SU}(2n)$. New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space $\mathbb{C}^n\simeq\mathbb{R}^{2n}$ underlies both global models, it is seen that for both systems the action variables generate the standard torus action on $\mathbb{C}^n$, and the fixe…
The three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is shown to be exactly solvable. When written in appropriate variables, its eigenfunctions can be expressed in terms of Jack symmetric polynomials. The exact solvability of the problem is explained by a hidden $sl(3,R)$ symmetry. A generalized Sutherland three-particle problem including both two- and three-body trigonometric potentials and internal degrees of freedom is then considered. It is analyzed in terms of three first-order noncommutin…
This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle duality for the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 3. We derive a Poisson-Lie deformation of the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 4. We construct a Lax pair for the hyperbolic Ruijsenaars-Schneider system with two couplings. 5. We present an explicit construction of compactified trigonometric and e…
We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of GL(n,C), which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard GL(n,C) Poisson--Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.
We apply a method of perturbation for the $BC_1$ Inozemtsev model from the trigonometric model and show the holomorphy of perturbation.Consequently, the convergence of eigenvalues and eigenfuncions which are expressed as formal power series is proved. We investigate also the relationship between $L^2$ space and some finite dimensional space of elliptic functions.
The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in $n+m$ variables, which reduce to the Jack polynomials when $n=0$ or $m=0$ and provide joint eigenfunctions of the quantum integrals of the trigonometric deformed Calogero-Moser-Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form $(p,q)\mapsto (L_pq)(0)$, with $L_p$ quantum integrals of the rational deformed Calogero-Moser-Sutherland system. In addition, we provide a new proof of the Lassalle-Nekrasov correspondence between trigonometric …
A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The ref…
The equilibrium positions of the multi-particle classical Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials associated with the classical root systems are described by the classical orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The eigenfunctions of the corresponding single-particle quantum CSM systems are also expressed in terms of the same orthogonal polynomials. We show that this interesting property is inherited by the Ruijsenaars-Schneider-van Diejen (RSvD) systems, which are integrable deformation of the CSM systems; the equilibrium po…
We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials $SP_λ((z_1,\ldots,z_n),(w_1,\ldots,w_m);θ)$ with respect to a natural positive semi-definite, but degenerate, Hermitian product $\langle\cdot,\cdot\rangle_{n,m}^\prime$. In case $m=0$ (or $n=0$), our product reduces to Macdonald's well-known inner product $\langle\cdot,\cdot\rangle_n^\prime$, and we recover his corresponding orthogonality results for the Jack polynomials $P_λ((z_1,\ldots,z_n);θ)$. From our main results, we readily infer that the kernel of $\langle\cdot,\cdot\rangle_{n,m}^\prime$ is s…
We consider a special class of quantum non-dynamical $R$-matrices in the fundamental representation of ${\rm GL}_N$ with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case $N=2$ these are the well-known 6-vertex $R$-matrix and its 7-vertex deformation. The $R$-matrices are used for construction of the classical relativistic integrable tops of the Euler-Arnold type. Namely, we describe the Lax pairs with spectral parameter, the inertia tensors and the Poisson structures. The latter are given by the linear Poisson-Lie brackets for th…
The wave functions of quantum Calogero-Sutherland systems for trigonometric case are related to polynomials in l variables (l is a rank of root system) and they are the generalization of Gegenbauer polynomials and Jack polynomials. Using the technique of κ-deformation of Clebsch-Gordan series developed in previous authors papers we investigate some new properties of generalized Gegenbauer polynomials.Note that similar results are also valid in A_2 case for more general two-parameter deformation ((q,t)-deformation) introduced by Macdonald.
In this paper we study the su(m) spin Sutherland (trigonometric) model of D_N type and its related spin chain of Haldane-Shastry type obtained by means of Polychronakos's freezing trick. As in the rational case recently studied by the authors, we show that these are new models, whose properties cannot be simply deduced from those of their well-known BC_N counterparts by taking a suitable limit. We identify the Weyl-invariant extended configuration space of the spin dynamical model, which turns out to be the N-dimensional generalization of a rhombic dodecahedron. This is in fact one of the reas…
Integrable deformations of the hyperbolic and trigonometric ${\mathrm{BC}}_n$ Sutherland models were recently derived via Hamiltonian reduction of certain free systems on the Heisenberg doubles of ${\mathrm{SU}}(n,n)$ and ${\mathrm{SU}}(2n)$, respectively. As a step towards constructing action-angle variables for these models, we here apply the same reduction to a different free system on the double of ${\mathrm{SU}}(2n)$ and thereby obtain a novel integrable many-body model of Ruijsenaars--Schneider--van Diejen type that is in action-angle duality with the respective deformed Sutherland model…
We compute the spectrum of the trigonometric Sutherland spin model of BC_N type in the presence of a constant magnetic field. Using Polychronakos's freezing trick, we derive an exact formula for the partition function of its associated Haldane-Shastry spin chain.
We present an algebraic construction of the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement. These eigenfunctions are the superspace extension of the generalized Hermite (or Hi-Jack) polynomials. The conserved quantities of the rational supersymmetric model are related to their trigonometric relatives through a similarity transformation. This leads to a simple expression between the corresponding eigenfunctions: the generalized Hermite superpolynomials are written as a differential operator acting on the corres…
We consider generalizations of the $BC$-type relativistic Calogero-Moser-Sutherland models, comprising of the rational, trigonometric, hyperbolic, and elliptic cases, due to Koornwinder and van Diejen, and construct an explicit eigenfunction for these generalizations. In special cases, we find the various kernel function identities, and also a Chalykh-Feigin-Sergeev-Veselov type deformation of these operators and their corresponding kernel functions, which generalize the known kernel functions for the Koornwinder-van Diejen models.
Certain aspects of the integrability/solvability of the Calogero-Sutherland-Moser systems and the Ruijsenaars-Schneider-van Diejen systems with rational and trigonometric potentials are reviewed. The equilibrium positions of classical multi-particle systems and the eigenfunctions of single-particle quantum mechanics are described by the same orthogonal polynomials: the Hermite, Laguerre, Jacobi, continuous Hahn, Wilson and Askey-Wilson polynomials. The Hamiltonians of these single-particle quantum mechanical systems have two remarkable properties, factorization and shape invariance.
We make a generalization of the type C monomial space of a single variable, which was introduced in the construction of type C N-fold supersymmetry, to several variables. Then, we construct the most general quasi-solvable second-order operators preserving this multivariate type C space. These operators of several variables are characterized by the fact that two different polynomial type solutions are available. In particular, we investigate and classify all the possible Schroedinger operators realized as a subclass of this family. It turns out that the rational, hyperbolic, and trigonometric C…
