the regular and singular Sturm-Liouville cases

A road can be smooth and well-marked the whole way, or it can run off the edge of the map or hit a washed-out bridge at one end. Sturm-Liouville problems split the same way. The regular ones are tame and well-behaved everywhere on a finite stretch; the singular ones misbehave at an endpoint or stretch out to infinity, and they need more care even though they are often the more physically interesting cases.

A Sturm-Liouville problem (p y')' + (q + lambda w) y = 0 on a <= x <= b is called regular when three things hold: the interval is finite, the functions p and w are strictly positive throughout (including at the ends), and the boundary conditions are the standard separated kind, one tidy condition at each end. Under these conditions the full theory holds in its cleanest form. It is called singular when any of these fails — for example p(x) or w(x) vanishes at an endpoint, or a coefficient blows up there, or the interval is infinite (like 0 to infinity or all of the real line).

The distinction is not pedantic, because the most important physical equations are singular. Bessel's equation has p(x) = x, which is zero at x = 0; Legendre's has p(x) = 1 - x^2, which vanishes at x = +1 and -1; the quantum harmonic oscillator lives on the whole infinite line. In these singular problems the explicit boundary condition at the bad point is often replaced by a softer requirement — that the solution merely stay bounded, or stay square-integrable, there. Remarkably, much of the regular theory survives: eigenvalues are still real, eigenfunctions still orthogonal. But surprises appear too, including spectra that become continuous rather than a discrete list.