41 research outputs found
Multivariable Bessel polynomials related to the hyperbolic Sutherland model with external Morse potential
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.Comment: a few minor misprints correcte
An Explicit Formula for Symmetric Polynomials Related to the Eigenfunctions of Calogero-Sutherland Models
We review a recent construction of an explicit analytic series representation
for symmetric polynomials which up to a groundstate factor are eigenfunctions
of Calogero-Sutherland type models. We also indicate a generalisation of this
result to polynomials which give the eigenfunctions of so-called 'deformed'
Calogero-Sutherland type models.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type
We introduce and study natural generalisations of the Hermite and Laguerre
polynomials in the ring of symmetric functions as eigenfunctions of
infinite-dimensional analogues of partial differential operators of
Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating
functions, duality relations, limit transitions from Jacobi symmetric
functions, and Pieri formulae, as well as the integrability of the
corresponding operators. We also determine all ideals in the ring of symmetric
functions that are spanned by either Hermite or Laguerre symmetric functions,
and by restriction of the corresponding infinite-dimensional CMS operators onto
quotient rings given by such ideals we obtain so-called deformed CMS operators.
As a consequence of this restriction procedure, we deduce, in particular,
infinite sets of polynomial eigenfunctions, which we shall refer to as super
Hermite and super Laguerre polynomials, as well as the integrability, of these
deformed CMS operators. We also introduce and study series of a generalised
hypergeometric type, in the context of both symmetric functions and 'super'
polynomials
A product formula for the eigenfunctions of a quartic oscillator
We consider the Schr\"odinger operator on the real line with an even quartic
potential. Our main result is a product formula of the type for its eigenfunctions
. The kernel function is given explicitly in terms of the
Airy function , and is positive for appropriate parameter
values. As an application, we obtain a particular asymptotic expansion of the
eigenfunctions .Comment: 18 pages. In v2 we added five references, reorganised some of the
material and made some minor revisions and corrections; and in v3 we added
references to work by T. T. Truong, who obtained a product formula for
quartic oscillator eigenfunctions already in 197
Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics
In the two preceding parts of this series of papers, we introduced and
studied a recursion scheme for constructing joint eigenfunctions of the Hamiltonians arising in the integrable -particle systems
of hyperbolic relativistic Calogero-Moser type. We focused on the first steps
of the scheme in Part I, and on the cases and in Part II. In this
paper, we determine the dominant asymptotics of a similarity transformed
function \rE_N(b;x,y) for , , and
thereby confirm the long standing conjecture that the particles in the
hyperbolic relativistic Calogero-Moser system exhibit soliton scattering. This
result generalizes a main result in Part II to all particle numbers .Comment: 21 page
A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type
In this paper we consider a large class of many-variable polynomials which
contains generalisations of the classical Hermite, Laguerre, Jacobi and Bessel
polynomials as special cases, and which occur as the polynomial part in the
eigenfunctions of Calogero-Sutherland type operators and their deformations
recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present
a unified and explicit construction of all these polynomials
Exact solutions of two complementary 1D quantum many-body systems on the half-line
We consider two particular 1D quantum many-body systems with local
interactions related to the root system . Both models describe identical
particles moving on the half-line with non-trivial boundary conditions at the
origin, and they are in many ways complementary to each other. We discuss the
Bethe Ansatz solution for the first model where the interaction potentials are
delta-functions, and we find that this provides an exact solution not only in
the boson case but even for the generalized model where the particles are
distinguishable. In the second model the particles have particular momentum
dependent interactions, and we find that it is non-trivial and exactly solvable
by Bethe Ansatz only in case the particles are fermions. This latter model has
a natural physical interpretation as the non-relativistic limit of the massive
Thirring model on the half-line. We establish a duality relation between the
bosonic delta-interaction model and the fermionic model with local momentum
dependent interactions. We also elaborate on the physical interpretation of
these models. In our discussion the Yang-Baxter relations and the Reflection
equation play a central role.Comment: 15 pages, a mistake corrected changing one of our conclusion