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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.

Interpret the Dirac delta coefficient as a point interaction in a second-order differential equation. The solution is continuous at $x=1$ and is twice differentiable on either side, where $y^{\prime \prime }=f$. Its transmission conditions for a delta coefficient areIndeed, the distributional second derivative across a jump contributes $[y^{\prime }{]}_1\delta (x-1)$, which must cancel the delta coefficient because $f$ has no point source.

Put $s=y^{\prime }(0)$ and $F(x)={\int }_0^x(x-t)f(t)dt$. Integrating the ordinary equation and inserting the required derivative jump giveswhere $(x-1{)}_{+}=\max (x-1,0)$. This formula is continuous, has the stated jump, satisfies the equation, and parametrizes every solution with the left endpoint value. Imposing the right endpoint value gives the scalar equationConsequently a unique $s$, and hence a unique solution for every $f,c_1,c_2$, exists precisely whenAt the exceptional value, the nonzero homogeneous solution with zero endpoint values isIt has derivative jump $-\pi /(\pi -1)=-{\alpha }_{\ast }\varphi (1)$. Thus uniqueness fails. More precisely, there are infinitely many solutions ifand none otherwise. This follows either from the scalar equation or by integration by parts against $\varphi$. It is the compatibility obstruction at Dirichlet resonance for a point interaction.