 Dirichlet Green function for a point interaction

For the point interaction in a second-order differential equation $L_{\alpha }={\partial }_x^2+\alpha {\delta }_c$ on $(0,\ell )$, use homogeneous Dirichlet boundary conditions. The unperturbed Green function iswith a positive unit derivative jump at $x=\xi$. Whenever $1+\alpha G_0(c,c)\ne 0$,To derive this rank-one operator correction, move the delta coefficient to the right-hand side and apply the unperturbed inverse: $G_{\alpha }(x,\xi )=G_0(x,\xi )-\alpha G_0(x,c)G_{\alpha }(c,\xi )$. Setting $x=c$ determines $G_{\alpha }(c,\xi )$ and gives the formula. Its denominator vanishes exactly at Dirichlet resonance for a point interaction.

The kernel is symmetric. It is continuous at $x=c$ and $x=\xi$, satisfies $[{\partial }_x{G}_{\alpha }{]}_{\xi }=1$ when $\xi \ne c$, and satisfies $[{\partial }_x{G}_{\alpha }{]}_c=-\alpha G_{\alpha }(c,\xi )$ away from the source. When $\xi =c$, the single combined condition is $[{\partial }_x{G}_{\alpha }{]}_c=1-\alpha G_{\alpha }(c,c)$. These are the transmission conditions for a delta coefficient, including a source at the interaction.