Depending on the dimension and the symmetry and form of the potential, the energy eigenvalues of a quantum mechanical system have different functional forms. Eg. The particle in the 1D-box gives rise to $E_n \propto n^2$, the hydrogen atom $E_n \propto -\frac{1}{n^2}$ the harmonic oscillator $E_n \propto n + \frac{1}{2}$, and so on. My question now is, could there possibly exist a system with energy eigenvalues $E_{n,m}\propto f(n m)$, for example like $\propto -\frac{1}{nm}$ with $n,m\in \mathbb{N}$?
I suspect the way the Hamilton function is translated to QM operators might prohibit such a product formation, since a product of say momenta ($\vec{p}$) is translated into a successive action of momentum operators rather than the product of the action of two operators. Which in turn will result in a summation of terms depending on different quantum numbers.
The question arises from thinking about how prime numbers could occur in physics. Since say, there would be a system with $E_{n,m} = \hbar(n+1)(m+1)$ with $n,m\in\mathbb{N}^+$ it would mean exactly non-prime number multiples of $\hbar$ would be allowed energy levels. And the prime numbers would appear as gaps in its eigenvalue spectrum.
