From Turbiner’s quasi-exactly soluble potentials in $N$ dimensions to analytic solutions of the combined Coulomb plus oscillator system

Turbiner’s one-dimensional, quasi-exactly soluble potentials can be interpreted as central potentials in $N$ dimensions. Using a particular transformation, one shows how these potentials convert to seemingly unrelated isolated solutions of other Schrödinger equations, including solutions of the combined Coulomb plus harmonic oscillator system.