Solvable PT -symmetric potentials in 2 and 3 dimensions

The solution of non-central PT -symmetric potentials is discussed by the separation of the variables in polar and angular coordinates. Conditions are formulated to guarantee the separation of the variables and the PT symmetry of the potential. The original eigenvalue equation is separated into one-dimensional Schrödinger-type differential equations. The importance of the boundary conditions, especially that of the periodic boundary condition of the azimuthal equation is pointed out. Further conditions leading to exact solutions of the whole problem are also formulated. An example combining the harmonic oscillator and the Scarf I potential in the radial and polar equation is discussed in detail, and the bound-state wave functions and the energy eigenvalues are derived. The spectrum exhibits partial degeneracies similar to those observed in the spectrum of the isotropic harmonic oscillator.