In my blog post Why riemannium? , I introduced the following idea. The infinite potential well in quantum mechanics, the harmonic oscillator and the Kepler (hydrogen-like) problem have energy spectra, respectively, equal to
1) $$ E\sim n^2$$ 2) $$ E\sim n$$ 3) $$ E\sim \dfrac{1}{n^2}$$
Do you know quantum systems with general spectra/eigenvalues given by
$$ E(n;s)\sim n^{-s}$$
and energy splitting
$$ \Delta E(n,m;s)\sim \left( \dfrac{1}{n^s}-\dfrac{1}{m^s}\right)$$
for all $s\neq -2,-1,2$?
Here we will not address the full quantum mechanical problem, but only discuss the semi-classical limit $n \gg 1$, i.e. only the highly excited part of the energy spectrum far away from the ground state energy.
If we are in one dimension with a power law potential
$$\Phi(x)~\sim~|x|^{p}, \qquad p>-2, $$
for $|x|$ sufficiently large, then we can use the semi-classical method of this Phys.SE answer to estimate the classically accessible length as
$$ \ell(V)~\sim~V^{\frac{1}{p}}, $$
where $V$ is the available potential energy. The number of states $N(E)$ below energy-level $E$ then goes as
$$ N(E)~\sim~E^{\frac{1}{p}+\frac{1}{2}}, $$
and therefore the semi-classical discrete energies also obey a power law
$$ E_n ~\sim~n^{\frac{2p}{p+2}} \quad\text{for}\quad n ~\gg~ 1. $$
The values $p=-1$, $p=2$, and $p=\infty$ correspond to the (radial) hydrogen atom, the harmonic oscillator, and the infinite potential well, respectively.
