 Boundary compatibility in the self-adjoint Fredholm alternative

For real $c$ and $L=\Delta +c$ on a bounded regular domain, let $N$ be the zero-boundary kernel. The problem $Lu=f$, $u=\psi$ on the boundary is solvable exactly when ${\int }_{\Omega }fz+{\int }_{\partial \Omega }\psi {\partial }_{\nu }z=0$ for every $z\in N$. The sign follows from Green second identity. After subtracting a boundary extension of $\psi$, this is the usual orthogonality of the forcing to the null space of the self-adjoint Dirichlet realization. All solutions then differ by an element of $N$.