Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials

As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of S U (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.