Algebraic description of S-N-invariant oscillator states

A group-theoretical description of N identical harmonic oscillators on a li ne is presented. It provides a scheme for labeling the energy eigenstates t hat are invariant under the permutation group S-N and for obtaining the sym metric operators that transform these degenerate eigenfunctions among thems elves. The symmetry algebra that these generators form is in general polyno mial. The 2- and 3- particle cases are considered in detail. For the simple 2- body problem the invariance algebra is found to be the cubic SU(2) alge bra: [J(0), J(+/-)] = +/-J(+/-), [J(+), J(-)] = 2J(0) - alpha J(0)(3.) In t he 3-body case, the permutational invariant states are characterized with t he help of the subgroup chain U(3) superset of U(2) superset of O(2). The l abeling and step operators are obtained from determining an integrity basis for the S-3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symm etry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided. (C) 1998 American Institute of Physics. [S00 22-2488(98)02007-6].