Classical integrable finite-dimensional systems related to Lie algebras

Systems of Calogero-Moser Type

“…Returning to the above free transforms, let us identify £ 2 ( G, dp) and L 2 (G,dx) via the antilinear map f(p) 1 -+ f(x). This is the quantum version of the identification of n and n explained in the paragraph containing (5.52): It entails that £8 and &a become antiunitary operators from L 2 ( G, dx) onto itself.…”

“…To handle the general case, it is convenient to introduce the vector The eigenfunctions E(x,p) can be written E(x,p) = A(x) 1 i.e., they admit multivariable polynomials as joint eigenfunctions. Taking f3 __, 0, this holds true for the correspondingly transformed PDOs A~"'' too (recall (4.45), (4.47), and (4.48) in this connection). '…”

“…A(x) 1 / 2 "" II sin Hxj -Xk)wr(Xj -xk) 1 We continue by explaining how the coefficients in (6.14) are determined_ To this end we introduce the renormalized polynomials Hence one obtains £0 = £8 , as announced above.…”

“…The systems just delineated were surveyed in the early eighties by 01-shanetsky and Perelomov, both in the classical [1] and in the quantum context [2]. These surveys contain extensive lists of references, and arc to a large extent concerned with the relations of the systemi:; to group theory, Lie algebra theory, and harmonic analysis on symmetric spaces.…”

Novel integrable spin-particle models from gauge theories on a cylinder

“…Most prominent examples, like the Kepler problem, are systems with few (≤ 3) degrees of freedom. An important exception is a class of integrable N-particle models associated with the names Calogero, Moser and Sutherland [1, 2] (for review see [3]). These are models for identical particles moving on one dimensional space and interacting via certain repulsive twobody potentials v(r).…”