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 $N$-body Hamiltonians, after appropriate gauge transformations, can be presented as a {\it quadratic} polynomial in the generators of the algebra $s{l}_N$ 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.

13 pages, latex. Connection to Jack polynomials clarified, misprints corrected

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