The one-dimensional harmonic oscillator in the presence of a dipole-like interaction

The Schrodinger equation for a one-dimensional harmonic oscillator is solved in the presence of an external dipole field. The solution is obtained in closed form and resembles the solution of the three-dimensional harmonic oscillator in spherical coordinates. Due to the existence of a singularity at the origin and the absence of tunneling, this one-dimensional system has a twofold degeneracy. Therefore, the nondegeneracy of the eigenvalues breaks down as has been observed in other systems.