The 2:1 anisotropic oscillator, separation of variables and symmetry group in Bargmann space

We present a detailed analysis of the separation of variables for the time-dependent Schrödinger equation for the anisotropic oscillator with a 2:1 frequency ratio. This reduces essentially to the time-independent one, where the known separability in Cartesian and parabolic coordinates applies. The eigenvalue problem in parabolic coordinates is a multiparameter one which is solved in a simple manner by transforming the system to Bargmann’s Hilbert space. There, the degeneracy space appears as a subspace of homogeneous polynomials which admit unique representations of a solvable symmetry algebra s3 in terms of first order operators. These representations, as well as their conjugate representations, are then integrated to indecomposable finite-dimensional nonunitary representations of the corresponding group S3. It is then shown that the two separable coordinate systems correspond to precisely the two orbits of the factor algebra s3/u (1) [u (1) generated by the Hamiltonian] under the adjoint action of the group. We derive some special function identitites for the new polynomials which occur in parabolic coordinates. The action of S3 induces a nonlinear canocical transformation in phase space which leaves the Hamiltonian invariant. We discuss the differences with previous works which present su (2) as the algebra responsible for the degeneracy of the two-dimensional anisotropic oscillator.