Exact Solvability of the Calogero and Sutherland Models

Abstract

Translationally invariant symmetric polynomials as coordinates for N-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland Nbody Hamiltonians, after appropriate gauge transformations, can be presented as a quadratic polynomial in the generators of the algebra sl N in finite-dimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed. a

Exact solutions of a new elliptic Calogero–Sutherland model

“…are the elementary symmetric functions of the variables z k [16]. When m is a non-negative integer, the generators (11) obviously preserve the finite-dimensional polynomial space…”

“…[16] and [17] by noting that the Hamiltonian can be mapped by a suitable gauge rotation to an element of the enveloping algebra of a certain realization of sl(N + 1) admitting finite-dimensional representations (cf. Eq.…”

Quasi-exactly Solvable (QES) Equations with Multiple Algebraisations

“…This number contrasts with the number of algebraic states existing in [5]. This gives a hint to the fact that further algebraisations might be possible to construct by introducing different "gauge factors".…”

“…Before addressing the main result of [4], we give the following definition of symmetric polynomials [5]:…”

“…This proposition results into the existence of seven possible new algebraisations. The algebraisations studied in [5] correspond to the choice n f = 0. The choices n f = 1 or n f = 2 lead to three new algebraisations according to the possible permutation of the non-trivial factor.…”