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Some potentials have only finitely many bound states (the finite square and delta function are two good examples) Others have infinitely many bound states (for example the infinite square well and $1/r$ hydrogen atom potential). What is a condition that guarantees a potential will have infinitely many bound states?

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    $\begingroup$ Related question in 1D: physics.stackexchange.com/q/485632/2451 $\endgroup$ – Qmechanic Jul 26 '19 at 5:34
  • $\begingroup$ What do you mean by "the" condition? What makes you think that there is only one such condition? $\endgroup$ – Emilio Pisanty Jul 27 '19 at 13:15
  • $\begingroup$ You raise a good point. In fact I think it's likely there are many equivalent conditions. Editing to "a" condition instead. $\endgroup$ – user2944352 Jul 28 '19 at 5:41
  • $\begingroup$ That is still too broad of a question for this site's format. $\endgroup$ – Emilio Pisanty Jul 28 '19 at 14:44
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A partial answer: The first chapter of Chadan and Sabatier’s ‘Inverse problems in quantum scattering theory’ explicitly defines so-called regular potentials

...to avoid the occurrence of some unnecessary complications (infinitely bound states etc). [emphasis added]

If we consider the scattering of a particle of mass m off a central potential, regular potentials are those that satisfy

$$\int_b^\infty |V(r)|r\,\mathrm dr < \infty$$ for $b \geq 0.$

So this is sufficient to avoid an infinite number of bound states for central potentials.

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