Related papers: Vector-valued Jack Polynomials and Wavefunctions o…
We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the…
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…
Using a similarity transformation that maps the Calogero model into $N$ decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions. The…
We argue that Jack Littlewood-Richardson coefficients $g_{\mu\nu}^{\lambda}(\alpha)$ are specialisations of certain novel polynomials. For the triple of partitions $(\mu,\nu,\lambda)=(21,21,321)$, we prove the corresponding polynomial is…
We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of…
We develop an algebraic approach for finding the eigenfunctions of a large class of few and many-body Hamiltonians, in one and higher dimensions, having linear spectra. The method presented enables one to exactly map these interacting…
We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders in the nome of the elliptic functions, for arbitrary values of the (positive) coupling…
A method for the calculation of translationally invariant wave functions for systems of identical fermions with arbitrary potential of pair interaction is developed. It is based on the well-known result that the essential dynamic part of…
Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random partitions. We connect these random partitions to the Hall-Littlewood polynomials of symmetric function theory, and use this connection to give combinatorial proofs…
We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between…
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
Duality identities in random matrix theory for products and powers of characteristic polynomials, and for moments, are reviewed. The structure of a typical duality identity for the average of a positive integer power $k$ of the…
We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in…
The explicit formula for the superconformal singular vectors in the Neveu-Schwarz sector has been obtained recently, via its symmetric polynomial representation, as a sum of Jack superpolynomials. Here we present the analogous, but slightly…
Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied.…
The Jack polynomials P_\lambda^{(\alpha)} at \alpha=-(k+1)/(r-1) indexed by certain (k,r,N)-admissible partitions are known to span an ideal I^{(k,r)}_N of the space of symmetric functions in N variables. The ideal I^{(k,r)}_N is invariant…
The Dunkl Laplacian is used to define the Hamiltonian of a modified quantum harmonic oscillator, associated with any finite reflection group. The potential is a sum of the inverse squares of the linear functions whose zero sets are the…
The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…
In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions $J_N$ for quantum Hamiltonians of the hyperbolic relativistic $N$-particle Calogero--Moser system. For generic coupling values, they are…
Calogero-Sutherland models associated to the Weyl groups of type A and B with exchange terms included in the Hamiltonians systems have non-symmetric eigenfunctions, which are products of the ground state with members of a family of…