Sturm-Liouville theory
/ SHTURM lyoo-VEEL /
Why do so many different physics problems -- a vibrating string, a quantum well, heat in a rod -- all end up with a discrete ladder of allowed modes, each with a clean shape, and any signal expandable as a sum over them? Sturm-Liouville theory is the single framework that explains it. It is the general theory behind Fourier series and all their special-function cousins at once.
A Sturm-Liouville problem writes a second-order ODE in the self-adjoint form d/dx[p(x) dy/dx] + q(x) y + lambda w(x) y = 0 on an interval with suitable boundary conditions, where w(x) greater than 0 is a weight. The theory guarantees: the eigenvalues lambda_n are real and form an increasing infinite sequence; the eigenfunctions y_n are orthogonal with respect to the weight w (the integral of y_m y_n w dx = 0 for m not equal to n); and -- the payoff -- the eigenfunctions are complete, so any well-behaved function expands as f(x) = the sum of c_n y_n(x), with coefficients c_n found by the weighted inner product. The operator being self-adjoint (Hermitian) is exactly what forces real eigenvalues and orthogonality.
Fourier series (p = w = 1), Legendre polynomials, Bessel functions, and Hermite polynomials are all just particular Sturm-Liouville problems with different p, q, w -- that is why they all share orthogonality and completeness. Above all, the time-independent Schrodinger equation is a Sturm-Liouville problem in disguise: the reality of energy eigenvalues, the orthogonality of stationary states, and the ability to expand any state in energy eigenstates are not quantum accidents but Sturm-Liouville theorems.
The vibrating string, y'' + lambda y = 0 with y(0) = y(L) = 0, is the simplest Sturm-Liouville problem (p = w = 1, q = 0). Its eigenvalues lambda_n = (n pi/L)^2 give the harmonic overtones and its eigenfunctions sin(n pi x/L) are the Fourier sine basis.
The Fourier sine series is the most basic Sturm-Liouville problem.
Completeness and orthogonality hinge on the boundary conditions making the operator self-adjoint; drop or mismatch them and eigenvalues can go complex and the eigenfunctions no longer form a usable basis.