a regular Sturm-Liouville problem
/ shtoorm lyoo-VEEL /
The cleanest, most trustworthy version of the Sturm-Liouville problem — the one where every textbook theorem holds without footnotes. Think of it as the problem on a closed, finite, well-behaved interval with no nasty edges.
A Sturm-Liouville problem -(p u')' + q u = lambda w u on [a, b] is called regular when three things hold: the interval is finite (a and b are real numbers, not infinity); the coefficients are nice — p, p', q, w continuous on the closed interval with p > 0 and w > 0 everywhere including the endpoints; and the boundary conditions are separated, meaning one condition at a and one at b of Robin type, alpha_1 u(a) + alpha_2 u'(a) = 0 and beta_1 u(b) + beta_2 u'(b) = 0. Dirichlet (u = 0) and Neumann (u' = 0) conditions are the familiar special cases.
For a regular problem the full Sturm-Liouville theorem applies in its strongest form: infinitely many real, simple eigenvalues lambda_1 < lambda_2 < lambda_3 < ... going off to plus infinity, each with a one-dimensional space of eigenfunctions, and the eigenfunctions form a complete orthogonal basis with weight w. The n-th eigenfunction has exactly n-1 interior zeros. This is the gold-standard case; the periodic and singular variants relax one of these hypotheses and pay a price (e.g. periodic problems can have double eigenvalues).
Heat in a rod 0 <= x <= 1 with insulated ends gives -X'' = lambda X, X'(0) = X'(1) = 0: a regular S-L problem (p = w = 1, q = 0, Neumann conditions). Eigenvalues lambda_n = (n*pi)^2 for n = 0, 1, 2, ..., eigenfunctions cos(n*pi*x), each with n interior zeros — a textbook-perfect regular case.
Finite interval, positive p and w throughout, separated boundary conditions: a regular problem.
Regularity is about the hypotheses, not the difficulty of solving. A regular problem may have no closed-form eigenfunctions at all (if p, q, w are messy functions), yet the theorem still guarantees the eigenvalues, orthogonality, and completeness exist.