= Dirichlet Green function for a point interaction
{c}
{title2=$G_\alpha=G_0-\alpha G_0(\cdot,c)G_0(c,\cdot)/(1+\alpha G_0(c,c))$}
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> is
$$
G_0(x,\xi)=-\frac{\min(x,\xi)\bigl(\ell-\max(x,\xi)\bigr)}\ell,
$$
with a positive unit derivative jump at $x=\xi$. Whenever $1+\alpha G_0(c,c)\ne0$,
$$
G_\alpha(x,\xi)=G_0(x,\xi)-\frac{\alpha G_0(x,c)G_0(c,\xi)}{1+\alpha G_0(c,c)}.
$$
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 $[\partial_xG_\alpha]_\xi=1$ when $\xi\ne c$, and satisfies $[\partial_xG_\alpha]_c=-\alpha G_\alpha(c,\xi)$ away from the source. When $\xi=c$, the single combined condition is $[\partial_xG_\alpha]_c=1-\alpha G_\alpha(c,c)$. These are the <transmission conditions for a delta coefficient>, including a source at the interaction.
\Image[/past-exam-of-the-differential-equations-2-course-of-the-university-of-oxford/2017-point-interaction-green-function.png]
{title=Resonance and source jumps for a one-dimensional point interaction}
{description=An original example with an interaction at $c=1$ and endpoint $\ell=\pi$. The left panel compares homogeneous solutions with the same initial data; the right panel shows Green kernels at coefficient one. A source placed at the interaction combines both derivative-jump contributions.}
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