the Sturm-Liouville theorem
/ shtoorm lyoo-VEEL /
This is the crown jewel — the single theorem that makes separation of variables a rigorous method rather than a hopeful trick. It says a Sturm-Liouville operator behaves exactly like a symmetric matrix, only with infinitely many eigenvalues. If you remember one result from this field, remember this one.
For a regular Sturm-Liouville problem -(p u')' + q u = lambda w u, the theorem makes four promises. First, the eigenvalues are all real, and there are infinitely many forming an increasing sequence lambda_1 < lambda_2 < lambda_3 < ... with lambda_n going to plus infinity. Second, each eigenvalue is simple — its eigenfunction is unique up to a constant multiple. Third, eigenfunctions for different eigenvalues are orthogonal with respect to the weight w: integral from a to b of u_m u_n w dx = 0 when m is not n. Fourth, the eigenfunctions are complete — they form a basis, so any sufficiently nice function f can be expanded as a generalized Fourier series f = sum of c_n u_n, and the series converges in mean-square (L^2) to f.
Every one of these four facts mirrors a property of a real symmetric matrix A: real eigenvalues, orthogonal eigenvectors, and a complete eigenbasis (the finite-dimensional spectral theorem). The Sturm-Liouville theorem is the infinite-dimensional version for the operator L = -(d/dx)(p d/dx) + q. It is precisely why you may write a temperature, a vibration, or a potential as a series in the natural modes of its domain and trust the answer.
For -X'' = lambda X with X(0) = X(L) = 0, the theorem predicts: real eigenvalues lambda_n = (n*pi/L)^2 (all positive, increasing to infinity), each simple, eigenfunctions sin(n*pi*x/L) orthogonal under integral from 0 to L of sin(m*pi*x/L) sin(n*pi*x/L) dx = 0, and completeness — every f on [0, L] equals its Fourier sine series in the mean-square sense. All four claims, verified.
The four guarantees — real, simple, orthogonal, complete — for the simplest Sturm-Liouville problem.
The full theorem (especially simplicity of eigenvalues and the n-zeros property) is stated for regular problems. For periodic boundary conditions eigenvalues can be double (cosine and sine share an eigenvalue), and for singular problems completeness still holds but the spectrum may include a continuous part.