solvability condition
Some equations refuse to be solved unless the right-hand side cooperates. The plainest example: you can solve u'' = f on a circle (periodic boundary) only if the average of f is zero, because integrating u'' around the loop gives zero on the left, so the total source must balance. A solvability condition is exactly such a constraint — a test the source must pass before any solution can exist at all.
The general statement comes from the adjoint. When the homogeneous problem L u = 0 has a nontrivial solution, the inhomogeneous problem L u = f is on the boundary between having no solution and having infinitely many. The Fredholm alternative settles it: L u = f is solvable if and only if f is orthogonal to every solution v of the homogeneous adjoint problem L* v = 0, meaning the integral of f times v vanishes for each such v. That orthogonality requirement is the solvability condition. When it holds, solutions exist but are not unique — you can add any homogeneous solution. When it fails, no solution exists.
Solvability conditions are everywhere once eigenvalues enter. A forced oscillator driven exactly at resonance has no bounded steady solution because the forcing is not orthogonal to the resonant mode — that is a solvability condition violated, and physically it is why resonance makes amplitudes blow up. In perturbation theory and multiple-scale analysis, imposing the solvability condition at each order is precisely what removes secular (unbounded) terms and determines the slow evolution. The condition is also what you must respect to define a modified Green's function when an ordinary one fails to exist.
Solve u'' + pi^2 u = f on [0, 1] with u(0) = u(1) = 0. The homogeneous problem has the nonzero solution sin(pi x) (pi^2 is an eigenvalue), so a solution exists only if integral from 0 to 1 of f(x) sin(pi x) dx = 0 — the source must have no component along the resonant mode.
When zero is effectively an eigenvalue, the source must be orthogonal to the corresponding mode or no solution exists.
The solvability condition tests orthogonality against the adjoint homogeneous solutions, not against L's own homogeneous solutions. For a self-adjoint problem the two coincide, which is why the distinction is easy to forget — but for a non-self-adjoint operator using the wrong null space gives the wrong condition.