a singular Sturm-Liouville problem
/ shtoorm lyoo-VEEL /
The wilder cousin of the regular problem, where one of the safe hypotheses fails — and yet, remarkably, almost all the famous special functions of physics come from here. Whenever you separate variables in a curved coordinate system (disk, cylinder, sphere), the radial or angular equation turns out to be singular.
A Sturm-Liouville problem is singular when the interval is infinite (like 0 to infinity or the whole real line), or when p(x) vanishes at an endpoint (so the leading coefficient degenerates), or when q or w blows up there. At such an endpoint you can no longer impose an ordinary boundary condition; instead you demand boundedness (the solution stays finite) or square-integrability with weight w. That replacement condition is what selects the physically meaningful solutions and discards the ones that blow up.
The big payoff: the eigenfunctions of singular problems are the named special functions. Bessel's equation -(x J')' + (n^2/x) J = lambda x J on a disk gives Bessel functions, orthogonal with weight x. Legendre's equation -((1 - x^2) P')' = lambda P on [-1, 1] gives Legendre polynomials. Hermite's and Laguerre's equations on infinite intervals give the Hermite and Laguerre functions of the quantum oscillator and the hydrogen atom. The spectrum may even turn partly continuous (as for the free particle), which is exactly where the discrete-series picture breaks and integral transforms take over.
Legendre's equation -((1 - x^2) P')' = lambda P on [-1, 1] is singular because p(x) = 1 - x^2 vanishes at both endpoints x = plus/minus 1. There is no room for a boundary condition there; instead one demands that P stay bounded at x = plus/minus 1. Only lambda = n(n+1) gives bounded solutions — the Legendre polynomials P_n.
When p vanishes at an endpoint, 'boundedness' replaces the usual boundary condition and quantizes lambda.
A common misconception is that singular problems still always have a purely discrete spectrum. They need not: on an unbounded interval the spectrum can be partly or wholly continuous (the free Schrodinger operator is the classic example), and then an eigenfunction expansion becomes an integral transform rather than a series.