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Assume a standard one-particle, non-relativistic Hamiltonian of the form \begin{equation} H=\frac{p^2}{2m}+V(r) \end{equation} and denote its eigenvalues as $E_{n,\tau}$, where $n$ is the principal quantum number and $\tau$ represents all the other quantum numbers, if any.

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 quantum numbers? My intuition is that long-range potentials should allow smaller gaps (cf. the Coulomb potential, where the gap gets infinitely small as $n\to\infty$), but I haven't managed to express this quantitatively.

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  • $\begingroup$ Take a look at the hydrogen spectrum in the limit of Rydberg atoms. What's not quantitative about the formula? It tells you the exact spectrum. You could, of course, ask about spectral properties of operators with the above form where $V(r)$ has a certain decay characteristic, but that's really a question for mathematics, where it has been explored plenty. $\endgroup$ – CuriousOne Jul 23 '16 at 22:18
  • $\begingroup$ Related: physics.stackexchange.com/questions/268872/… (which has a link to a question about ::drumroll:: Rydberg atoms. $\endgroup$ – dmckee Jul 23 '16 at 22:20
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    $\begingroup$ That's a problem handled in functional analysis. You need to talk to the mathematicians about that. It is, in its general form, a complicated problem, by the way. There is no simple answer and the relevant answers lead you far away from the realm of $L^2$ functions, which are pretty much the only ones we care about in physics. $\endgroup$ – CuriousOne Jul 23 '16 at 22:24
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    $\begingroup$ It is not clear what you are asking. How can consecutive energy levels have the same quantum numbers? You have not defined the potential. Are you asking if there is a general result applying to all potentials? $\endgroup$ – sammy gerbil Jul 23 '16 at 22:28
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    $\begingroup$ @sammygerbil I find the problem statement perfectly clear. The OP is asking about holding all other quantum numbers fixed, such as the magnetic quantum number. $\endgroup$ – knzhou Jul 23 '16 at 22:31

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