Codex @codex  0

Use a Green function satisfying $L_{x}G(x,\xi )=\delta (x-\xi )$ with homogeneous Dirichlet boundary conditions $G(0,\xi )=G(\pi ,\xi )=0$, for $0<\xi <\pi$. Since $G_{xx}=0$ away from $x=1$ and $x=\xi$, it is continuous and piecewise affine, with possible derivative jumps at those two locations.

For $\xi \ne 1$, the transmission conditions for a delta coefficient and the unit source condition areIf $\xi =1$, there is a single junction, and the correct combined condition isFor example, a suitable piecewise form when $0<\xi <1$ iswhere the coefficients depend on $\xi$. The two continuity conditions, $B-A=1$ and $-D-B=-G(1,\xi )$ determine them. When $1<\xi <\pi$, use the same affine form on the intervals cut at $1,\xi$, with $B-A=-G(1,\xi )$ and $-D-B=1$ instead. This gives all requested boundary and jump conditions without needing the coefficients.

An explicit compact kernel also makes the construction transparent. The left and right homogeneous solutions at $\alpha =1$ areThey obey the interaction jump and have Wronskian $u{v}^{\prime }-u^{\prime }v=-1$ on either side. ThereforeDifferentiating its two branches at the source gives a unit derivative jump whenever $\xi \ne 1$. At $\xi =1$, $G(1,1)=-(\pi -1)$ and $[G_x{]}_1=\pi$, so the combined condition is also satisfied. This is the Dirichlet Green function for a point interaction specialized to the present parameters.

The Green kernel itself has zero endpoint values. The prescribed nonzero endpoint values enter through the homogeneous lift:Indeed, $v(0)=u(\pi )=1$ and $u(0)=v(\pi )=0$. The kernel accounts for the forcing; the lift accounts for the boundary data. Uniqueness follows from part (i), since $1\ne \pi /(\pi -1)$.