the Fredholm alternative
/ FRED-holm /
For a square matrix, you learned a clean dichotomy in linear algebra: either A x = b has a unique solution for every b, or A is singular — and then A x = b is solvable only for special b, namely those orthogonal to the solutions of the transposed homogeneous system. The Fredholm alternative is exactly this dichotomy, lifted from finite matrices to the infinite-dimensional world of elliptic boundary-value problems. It tells you when L u = f is solvable and how badly uniqueness can fail.
For an elliptic operator L on a bounded domain (say with Dirichlet conditions), the alternative says precisely one of two cases holds. Case one: the only solution of the homogeneous problem L u = 0 is u = 0. Then L u = f has a unique solution for every right-hand side f — the operator is invertible. Case two: L u = 0 has nontrivial solutions (a finite-dimensional space of them). Then L u = f is solvable if and only if f is orthogonal to every solution of the adjoint homogeneous problem L* v = 0 — and when a solution exists it is not unique, you may add any homogeneous solution. The reason it works is compactness: solving the elliptic equation is the same as inverting (I minus a compact operator), via the compact embedding of Sobolev spaces (Rellich-Kondrachov), and compact perturbations of the identity obey exactly the matrix dichotomy.
This is the organising principle for solvability with eigenvalues in play. Consider - Laplacian u - lambda u = f with Dirichlet data. For most lambda the homogeneous problem has only u = 0, so you can always solve. But if lambda equals a Dirichlet eigenvalue, the homogeneous problem has eigenfunctions, and now L u = f is solvable only when f is orthogonal to those eigenfunctions — the same resonance condition you meet for a forced oscillator at its natural frequency. The Fredholm alternative is what makes that resonance precise.
On (0, pi) with u(0) = u(pi) = 0, solve u'' + u = f. The homogeneous problem u'' + u = 0 has the nonzero solution sin x (since lambda = 1 is a Dirichlet eigenvalue). By the Fredholm alternative the equation is solvable only when the integral from 0 to pi of f(x) sin x dx = 0; otherwise there is no solution at all, and when there is one you may add any multiple of sin x.
At resonance solvability becomes conditional: the forcing must be orthogonal to the resonant mode.
The orthogonality condition is against the ADJOINT homogeneous solutions, not the original ones. For a self-adjoint operator (like the symmetric divergence-form case) the adjoint equals the operator itself, so the two coincide and the condition reads simply that f is orthogonal to the eigenfunctions — but for non-self-adjoint L you must use the adjoint.