Inequivalence of the classes of classical and quantum harmonic potentials: Proof by example

There are an infinite number of potentials whose classical angular velocity ωc is independent of energy. There are also an infinite number of potentials whose quantum eigenenergies are equally spaced, ωq. If ωc=ωq, then in the WKB approximation these classes are the same. (Certain specific potentials belong to both of these classes exactly.) However, we show by a different specific example that in general these classes are not equivalent.