The authors of this unreviewed article discuss the observation that the spectrum $E_n$ of the eigenvalues of the Hamiltonian $\hat{H}$ of a 1D quantum system is always of the form

$E_n = c_0 + c_1 n^{c_2},\hspace{20pt}c_j \in \mathbb{N}$,

when the potential energy is a monomial $V(x) = Ax^m$ with $m$ an even integer. When $m = 2$, it's clear that the constant $c_2$ has value $1$ because then it's a harmonic oscillator.

After calculating the first 10 or so energies for many different kinds of functions $V(x)$, it seems to me that actually the spectrum $E_n$ is of this form for any *convex function* $V(x)$, i.e. a potential with a graph that is turning upwards at any point $x$. This also holds for functions with a discontinuous derivative, such as $V(x) = A|x|$ and $V(x) = Ae^{|x|}$. When $V(x)$ is not convex, such as for a double well potential with a barrier in between, this does not hold. For instance, the most significant property of the 1D double well is that the spacing between the two lowest energy states is disproportionately small compared to other energy spacings, and therefore that power function doesn't fit the eigenvalues.

Has this been observed by anyone before, and does there exist a mathematical proof for it?