4
$\begingroup$

Why does Mother Nature allow bound states in arbitrarily weak attractive potential in 2D but not in 3D?

See, for example, this article, arXiv:math-ph/0208011.

$\endgroup$
  • 1
    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/143630/2451 $\endgroup$ – Qmechanic Sep 1 '14 at 18:45
  • $\begingroup$ Could you perhaps elaborate on what you mean when you ask Why? here? Are you asking what step of the proof breaks down in higher dimensions? Or are you asking if there are no-go theorems forbidding such bound states in higher dimensions? Or something else? $\endgroup$ – ACuriousMind Sep 1 '14 at 18:49
  • $\begingroup$ @ACuriousMind Sure. I'll address your questions one by one. First, I haven't seen a mathematical proof that the statement does not hold in 3D, so it would be great to see such a proof and particularly how it breaks down in higher dimensions. Second, I'm not aware of any such theorems, so it would also be very exciting to know. Third, is there an intuitive explanation, for example, why would higher dimensionality impairs that ability of space to support the bound states? $\endgroup$ – Taiben Sep 1 '14 at 18:55
  • $\begingroup$ @ACuriousMind When I asked this question, I was hoping to see an answer to the third aspect -- intuitive reasoning of the underlying physics. $\endgroup$ – Taiben Sep 1 '14 at 18:56
  • 1
    $\begingroup$ It is explained here (pages $159,160$), at least for a simple potential $U(r)= - u \delta(r)$ $\endgroup$ – Trimok Sep 2 '14 at 9:35