1,568 research outputs found
Explicit formulas for the generalized Hermite polynomials in superspace
We provide explicit formulas for the orthogonal eigenfunctions of the
supersymmetric extension of the rational Calogero-Moser-Sutherland model with
harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in
superspace. The construction relies on the triangular action of the Hamiltonian
on the supermonomial basis. This translates into determinantal expressions for
the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version
of hep-th/0305038 which has been splitted in two articles. In this revised
version, the introduction has been rewritten and a new appendix has been
added. To appear in JP
Rodrigues Formula for Hi-Jack Symmetric Polynomials Associated with the Quantum Calogero Model
The Hi-Jack symmetric polynomials, which are associated with the simultaneous
eigenstates for the first and second conserved operators of the quantum
Calogero model, are studied. Using the algebraic properties of the Dunkl
operators for the model, we derive the Rodrigues formula for the Hi-Jack
symmetric polynomials. Some properties of the Hi-Jack polynomials and the
relationships with the Jack symmetric polynomials and with the basis given by
the QISM approach are presented. The Hi-Jack symmetric polynomials are strong
candidates for the orthogonal basis of the quantum Calogero model.Comment: 17 pages, LaTeX file using jpsj.sty (ver. 0.8), cite.sty,
subeqna.sty, subeqn.sty, jpsjbs1.sty and jpsjbs2.sty (all included.) You can
get all the macros from ftp.u-tokyo.ac.jp/pub/SOCIETY/JPSJ
Shape Invariance in the Calogero and Calogero-Sutherland Models
We show that the Calogero and Calogero-Sutherland models possess an N-body
generalization of shape invariance. We obtain the operator representation that
gives rise to this result, and discuss the implications of this result,
including the possibility of solving these models using algebraic methods based
on this shape invariance. Our representation gives us a natural way to
construct supersymmetric generalizations of these models, which are interesting
both in their own right and for the insights they offer in connection with the
exact solubility of these models.Comment: Latex file, 23 pages, no picture
Rodrigues Formula for the Nonsymmetric Multivariable Hermite Polynomial
Applying a method developed by Takamura and Takano for the nonsymmetric Jack
polynomial, we present the Rodrigues formula for the nonsymmetric multivariable
Hermite polynomial.Comment: 5 pages, LaTe
Jack vertex operators and realization of Jack functions
We give an iterative method to realize general Jack functions from Jack
functions of rectangular shapes. We first show some cases of Stanley's
conjecture on positivity of the Littlewood-Richardson coefficients, and then
use this method to give a new realization of Jack functions. We also show in
general that vectors of products of Jack vertex operators form a basis of
symmetric functions. In particular this gives a new proof of linear
independence for the rectangular and marked rectangular Jack vertex operators.
Thirdly a generalized Frobenius formula for Jack functions was given and was
used to give new evaluation of Dyson integrals and even powers of Vandermonde
determinant.Comment: Expanded versio
Quantum Calogero-Moser Models: Integrability for all Root Systems
The issues related to the integrability of quantum Calogero-Moser models
based on any root systems are addressed. For the models with degenerate
potentials, i.e. the rational with/without the harmonic confining force, the
hyperbolic and the trigonometric, we demonstrate the following for all the root
systems: (i) Construction of a complete set of quantum conserved quantities in
terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal
R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the
Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of
the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack
polynomials are defined for all root systems as unique eigenfunctions of the
Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v)
Algebraic construction of all excited states in terms of creation operators.
These are mainly generalisations of the results known for the models based on
the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure
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=\infty, which lead to two families of Hall-Littlewood polynomials in
superspace. We also find that the Macdonald polynomials in superspace evaluated
at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In
particular, their expansion coefficients in the corresponding Hall-Littlewood
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.Comment: 18 page
Equivalence of the super Lax and local Dunkl operators for Calogero-like models
Following Shastry and Sutherland I construct the super Lax operators for the
Calogero model in the oscillator potential. These operators can be used for the
derivation of the eigenfunctions and integrals of motion of the Calogero model
and its supersymmetric version. They allow to infer several relations involving
the Lax matrices for this model in a fast way. It is shown that the super Lax
operators for the Calogero and Sutherland models can be expressed in terms of
the supercharges and so called local Dunkl operators constructed in our recent
paper with M. Ioffe. Several important relations involving Lax matrices and
Hamiltonians of the Calogero and Sutherland models are easily derived from the
properties of Dunkl operators.Comment: 25 pages, Latex, no figures. Accepted for publication in: Jounal of
Physics A: Mathematical and Genera
Collective Field Description of Spin Calogero-Sutherland Models
Using the collective field technique, we give the description of the spin
Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be
applicable for arbitrary coupling constant and provides the bosonized
Hamiltonian of the spin CSM. The boson Fock space can be identified with the
Hilbert space of the spin CSM in the large limit. We show that the
eigenstates corresponding to the Young diagram with a single row or column are
represented by the vertex operators. We also derive a dual description of the
Hamiltonian and comment on the construction of the general eigenstates.Comment: 14 pages, one figure, LaTeX, with minor correction
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