Jack Superpolynomials: Physical and Combinatorial Definitions

Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland Hamiltonian. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this quantum-mechanical problem. But Jack superpolynomials can also be defined more combinatorially, starting from the multiplicative bases of symmetric superpolynomials, enforcing orthogonality with respect to a one-parameter deformation of the combinatorial scalar product. Both constructions turn out to be equivalent.     

Solvability of F 4 Quantum Integrable Systems

It is shown that the F ₄ rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are obtained by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are a certain invariants of the F ₄ Weyl group. Both Hamiltonians preserve the same (minimal) flag of spaces of polynomials, which is found explicitly.     

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

Solvability of the rational quantum integrable systems related to exceptional root spaces G₂,F₄ is re-examined and for E_6,7,8 is established in the framework of a unified approach. It is shown that Hamiltonians take algebraic form being written in certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.Alexander V. Turbiner: On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia.     

Jack Superpolynomials with Negative Fractional Parameter: Clustering Properties and Super-Virasoro Ideals

The Jack polynomials ${P_\lambda^{(\alpha)}}$ at ???= ?(k?+?1)/(r ? 1) indexed by certain (k, r, N)-admissible partitions are known to span an ideal ${I_{N}^{(k,r)}}$ of the space of symmetric functions in N variables. The ideal ${I_{N}^{(k,r)}}$ is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in ${I_{N}^{(k,r)}}$ admit clusters of size at most k: they vanish when k?+?1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials ${P_\lambda^{(\alpha)}}$ at ???= ?(k?+?1)/(r ? 1) indexed by certain (k, r, N)-admissible superpartitions span an ideal ${\mathcal{I}_{N}^{(k,r)}}$ of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal ${\mathcal{I}_{N}^{(k,r)}}$ is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in ${\mathcal {I}_{N}^{(k,r)}}$ vanish when k?+?1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of ${\mathcal{I}_{N}^{(k,2)}}$ provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k?+?1 commuting variables are set equal to each other.     

Jack Superpolynomials,Superpartition Ordering and Determinantal Formulas

 We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the model's Hamiltonian on monomial superfunctions. This allows to define Jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly, with respect to this ordering, on the basis of monomial superfunctions. This further leads to a simple and explicit determinantal expression for the Jack superpolynomials. Received: 11 May 2001 / Accepted: 16 August 2002 Published online: 7 November 2002 Communicated by R. H. Dijkgraaf     

Macdonald Polynomials in Superspace: Conjectural Definition and Positivity Conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace and a rather non-trivial expression for their evaluation. We study the limiting cases q =?0 and q =???, which lead to two families of Hall?CLittlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q =?t =?0 or q =?t = ?? seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall?CLittlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q = t =?0 family) are polynomials in q and t with nonnegative integer coefficients.     

Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials

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 reformulation of the models in terms of the exchange-operator formalism is a crucial aspect of our analysis.     

Infinitely many shape invariant potentials and new orthogonal polynomials

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Pöschl–Teller potentials in terms of their degree ℓ polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (ℓ=1,2,…) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and Gómez-Ullate et al.'s are the first members of these infinitely many potentials.     

An algebraic approach to the non-symmetric Macdonald polynomial

In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are simultaneous eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them.     

The Calogero-Sutherland model and polynomials with prescribed symmetry

The Schrödinger operators with exchange terms for certain Calogero-Sutherland quantum many-body systems have eigenfunctions which factor into the symmetric ground state and a multivariable polynomial. The polynomial can be chosen to have a prescribed symmetry (i.e. be symmetric or antisymmetric) with respect to the interchange of some specified variables. For four particular Calogero-Sutherland systems we construct an eigenoperator for these polynomials which separates the eigenvalues and establishes orthogonality. In two of the cases this involves identifying new operators which commute with the corresponding Schrödinger operators. In each case we express a particular class of the polynomials with prescribed symmetry in a factored form involving the corresponding symmetric polynomials.     

Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle

We introduce an integrable lattice discretization of the quantum system of n bosonic particles on a ring interacting pairwise via repulsive delta potentials. The corresponding (finite-dimensional) spectral problem of the integrable lattice model is solved by means of the Bethe Ansatz method. The resulting eigenfunctions turn out to be given by specializations of the Hall-Littlewood polynomials. In the continuum limit the solution of the repulsive delta Bose gas due to Lieb and Liniger is recovered, including the orthogonality of the Bethe wave functions first proved by Dorlas (extending previous work of C.N. Yang and C.P. Yang).Work supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grant # 1051012, by the Anillo Ecuaciones Asociadas a Reticulados financed by the World Bank through the Programa Bicentenario de Ciencia y Tecnología, and by the Programa Reticulados y Ecuaciones of the Universidad de Talca.     

Sekiguchi-Debiard Operators at Infinity

We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables ${x_1,x_2, \ldots}$ are their eigenfunctions. These operators are defined as limits at N → ∞ of renormalised Sekiguchi-Debiard operators acting on symmetric polynomials in the variables ${x_1 , \ldots , x_N}$ . They are differential operators in terms of the power sum variables ${p_n=x_1^n+x_2^n+\cdots}$ and we compute their symbols by using the Jack reproducing kernel. Our result yields a hierarchy of commuting Hamiltonians for the quantum Calogero-Sutherland model with an infinite number of bosonic particles in terms of the collective variables of the model. Our result also yields the elementary step operators for the Jack symmetric functions.     

Romanovski polynomials in selected physics problems

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.     

Residual gauge invariance of Hamiltonian lattice gauge theories

The time-independent residual gauge invariance of Hamiltonian lattice gauge theories is considered. Eigenvalues and eigenfunctions of the unperturbed Hamiltonian are found in terms of Gegenbauer's polynomials. Physical states which satisfy the subsidiary condition corresponding to Gauss' law are constructed systematically.     

Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model

A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.     

Szegő Difference Equations, Transfer Matrices¶and Orthogonal Polynomials on the Unit Circle

We develop the theory of orthogonal polynomials on the unit circle based on the Szegő recurrence relations written in matrix form. The orthogonality measure and C-function arise in exactly the same way as Weyl's function in the Weyl approach to second order linear differential equations on the half-line. The main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one-dimensional Schr?dinger operators (discrete and continuous), and the notion of subordinacy from the Gilbert–Pearson theory. We study the relations between transfer matrices and the structure of orthogonality measures. The theory is illustrated by the Szegő equations with reflection coefficients having bounded variation. Received: 26 February 2001 / Accepted: 28 May 2001     

Orthogonal harmonic polynomials onU(2)

Electric charges and free electromagnetic waves are supposed to be described locally with the same wave differential equation. It is only the topology that is considered to be different. The calculated nonlocalU(2) individuals are characterized by polynomials that belong neither to the classical nor to the Szegö polynomials. The construction of the polynomial solution in component form, their orthogonality over singular measures, the relationships to the Jacobi polynomials, Rodriguez formulas, product decomposition, asymptotic formulas, and completeness are presented in some detail. The possibility is discussed of whether this highly nonlocal model for electric charges can have a physical significance. This work is intended to be a first step for the realization of an old idea of Einstein's (and also commented on by Dirac) to start with the electric charge, not with the Planck constant, as the primary concept for quantum theory.     

Anyons and the Elliptic Calogero–Sutherland Model

We obtain a second quantization of the elliptic Calogero–Sutherland (eCS) model by constructing a quantum field theory model of anyons on a circle and at a finite temperature. This yields a remarkable identity involving anyon correlation functions and providing an algorithm for solving of the eCS model. The eigenfunctions obtained define an elliptic generalization of the Jack polynomials.     

Confluent Hypergeometric Orthogonal Polynomials Related to the Rational Quantum Calogero System with Harmonic Confinement

Two families (type A and type B) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri-type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type A) and the Laguerre polynomials (type B), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl. Received:Received: 20 October 1996 / Accepted: 3 March 1997     

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.Research supported in part by NSF Grant No. MCS-76-05857Research supported in part by NSF Grant No. MCS-74-07313-A02