A similar version of this question was originally posted by me in the physics community, but it was suggested that I ask the mathematicians instead. So I have tried to strip off most of the physics jargon and brush up my functional analysis-fu.
Consider the stationary Schrödinger equation on $L_2(\mathbb{R}^3)$: \begin{equation} -\nabla^2\psi(\vec r)+V(\vec r)\psi(\vec r) = E\psi(\vec r)\text. \end{equation} Here $V(\vec r)$ can be assumed to be a sufficiently well-behaved function that vanishes at infinity. Assume this operator has a discrete spectrum and denote the discrete eigenvalues as $E_{n,\tau}$, where $n$ is the principal quantum number and $\tau$ represents all the other quantum numbers, if any.
For a given $V(\vec r)$, is it possible to derive a lower bound on $E_{n+1,\tau}-E_{n,\tau}$, i.e. on the energy gap between consecutive levels with the same additional quantum numbers? Physical intuition tells me that long-range potentials should allow smaller gaps (cf. the Coulomb potential $V(r)=1/r$, which yields $E_n\propto 1/n^2$, with the gap getting infinitely small as $n\to\infty$), but I haven't managed to express this quantitatively.