Yangian Gelfand-Zetlin bases, \({\mathfrak gl}_N\)-Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model
We consider the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland models with \(N= 1,2,3,\dots\) in the framework of symmetric polynomals. In this framework it becomes apparent that all these models are manifestations of the same entity, which is the commuting family of Macdonald operators. Macdonald operators depend on two parameters \(q\) and \(t\). The Hamiltonian of the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland model belongs to a degeneration of this family in the limit when both \(q\) and \(t\) approach an \(N\)th elementary root of unity. This is a generalization of the well-known situation in the case of the scalar Calogero-Sutherland model \((N=1)\). In the limit the commuting family of Macdonald operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland model. The limits of Macdonald polynomials which we call \({\mathfrak {gl}}_N\)-Jack polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The \({\mathfrak {gl}}_N\)-Jack polynomials describe the orthogonal eigenbasis of the \({\mathfrak {gl}}_N\)-invariant Calogero-Sutherland model in exactly the same way as Jack polynomials describe the orthogonal eigenbasis of the scalar model \((N=1)\). For each known property of Macdonald polynomials there is a corresponding property of \({\mathfrak {gl}}_N\)-Jack polynomials. As a simplest application of these properties we compute two-point dynamical spin-density and density correlation functions in the \({\mathfrak {gl}}_2\)-invariant Calogero-Sutherland model at integer values of the coupling constant.
scientific article