Quantization of the charge in Coulomb plus harmonic potential

Abstract

We consider two models where the wave equation can be reduced to the effective Schrödinger equation whose potential contains both harmonic and the Coulomb terms, ω2r2−a/r⁠. The equation reduces to the biconfluent Heun’s equation, and we find that the charge as well as the energy must be quantized and state-dependent. We also find that two quantum numbers are necessary to count radial degrees of freedom and suggest that this is a general feature of differential equation with higher singularity like the Heun’s equation.