I am currently dealing with scattering theory. I looked up the scattering on a spherical well potential.

$$V(r) = \begin{cases} -V_0 & , r \leq R\\ 0 & ,r > R \end{cases} $$

where $V_0 > 0$ is the potential depth and $R$ corresponds to the potential width. Since I look only at s-waves, we can set $l=0$. I calculated then the scattering length $$a \approx R(1-\frac{\tan(\gamma)}{\gamma}), $$ with $$\gamma = \frac{R\sqrt{2mV_0}}{\hbar}. $$ Now $a$ diverges for values $\gamma = (n+\frac{1}{2})\pi$. Now I read somewhere without explanation that this exactly the condition for $n$ bound states $(E<0)$ and one metastable bound state $(E=0)$ in a spherical potential well. Looking at the solution of the spherical well, I see the bound states, but I dont see what is meant with those metastable states. I would be very glad if somebody could explain this to me.


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