Eigenvalue Problems and Sturm–Liouville Theory
Stub. Two-point boundary value (endpoint) problems and their eigenvalues/eigenfunctions, worked examples, and general Sturm–Liouville theory — the framework that explains why eigenfunctions are orthogonal and form a complete basis.
Replaces an initial-value problem with a boundary-value (endpoint) problem, where nontrivial solutions exist only for special values of a parameter — the eigenvalues, with associated eigenfunctions. Works through examples before stating the general theory: a Sturm–Liouville problem is the abstract structure ( with boundary conditions) that guarantees real eigenvalues and mutually orthogonal eigenfunctions, which is exactly why separation of variables for the heat/wave/Laplace equations produces a Fourier-like eigenfunction expansion rather than an ad hoc one.
Outline (TODO — flesh out each)
- Endpoint problems and eigenvalues
- More examples of eigenvalue problems
- Sturm–Liouville theory I — the general framework, self-adjointness
- Sturm–Liouville theory II — orthogonality of eigenfunctions, eigenfunction expansions