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

Publisher Website
Google Scholar
Abstract
We present a detailed analysis of the separation of variables for the time‐dependent Schrödinger equation for the anisotropicoscillator 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 algebras 3 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 S 3. It is then shown that the two separable coordinate systems correspond to precisely the two orbits of the factor algebras 3/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 S 3 induces a nonlinear canocical transformation in phase space which leaves the Hamiltonian invariant. We discuss the differences with previous works which present s u (2) as the algebra responsible for the degeneracy of the two‐dimensional anisotropicoscillator.