Abstract
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, $\omega^{2}r^{2}-a/r$. 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.
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