# Boundary condition Hamiltonian with point tinteractions

I`m studying the Hamiltonian with point interaction centered in $y$ in three dimensions. I know that the elements in the domain of the Hamiltonian are of the form $$\psi=\phi+qG^z(\cdot-y)$$ where $G^z(x)=\frac{e^{i\sqrt{z}|x|}}{4\pi|x|}$, $q\in\mathbb{C}$ and $\phi\in H^2$. The boundary condition is $$q=\frac{4\pi\phi(y)}{4\pi\alpha-i\sqrt{z}}$$ Many times it is written in the form $$\lim_{r\to 0}\bigg[\frac{\partial (r\psi)}{\partial r}-4\pi\alpha(r\psi)\bigg]=0$$. Are these two relations equivalent?

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There are some misprints in your post. To get replies, you should be a lot more explicit, I guess. –  Vibert Apr 11 '13 at 17:52
What misprints? –  Math Apr 11 '13 at 19:31