Differential Equations

Sturm–Liouville Theory

The structure shared by every well-posed boundary value problem. Its orthogonal basis is not an accident.

Updated May 2026

Fourier series, Legendre polynomials, Bessel functions, spherical harmonics, Hermite polynomials. These appear in unrelated corners of physics and all have the same two properties: they are orthogonal, and they are complete. Sturm-Liouville theory is where that shared structure comes from.

The general form

\frac{d}{d x} [p (x) \frac{d y}{d x}] + q (x) y + λ w (x) y = 0

on $[a, b]$ with $p, w > 0$ and appropriate boundary conditions. Every classical special function of physics is an eigenfunction of some Sturm–Liouville problem.

Theorem

Sturm–Liouville

For a regular problem:

The eigenvalues are real, and form an increasing sequence $λ_{1} < λ_{2} < \dots \to \infty$ .
Eigenfunctions for distinct eigenvalues are orthogonal with respect to the weight $w$ : $\int_{a}^{b} y_{m} (x) y_{n} (x) w (x) d x = 0, m \neq = n$
The eigenfunctions are complete: any sufficiently well-behaved function on $[a, b]$ expands in them.
$y_{n}$ has exactly $n - 1$ interior zeros.

Why orthogonality is automatic

The operator is self-adjoint. Integrate by parts twice and the boundary terms cancel by the boundary conditions, leaving

(λ_{m} - λ_{n}) \int_{a}^{b} y_{m} y_{n} w d x = 0

so distinct eigenvalues force orthogonality. Nothing about the specific $p$ , $q$ or $w$ entered. Self-adjointness is the whole reason, which is also why Hermitian operators are non-negotiable in quantum mechanics: they are what guarantee real eigenvalues and an orthogonal basis to measure in.

The zeros, and why they matter

Property 4 is the least famous and often the most useful. The ground state has no interior zeros; the $n$ -th excited state has $n - 1$ .

In quantum mechanics this is the node-counting theorem: count the nodes in a wavefunction and you know which level it is, without solving anything. It also underpins the shooting method. Sweep the energy, count nodes, and you know exactly which way to move.

The catalogue

Problem	Interval	Weight	Eigenfunctions
$y^{''} + λ y = 0$	$[0, L]$	$1$	$sin (nπ x / L)$
Legendre	$[- 1, 1]$	$1$	$P_{ℓ} (x)$
Bessel	$[0, a]$	$x$	$J_{m} (α_{mn} x / a)$
Hermite	$(- \infty, \infty)$	$e^{- x^{2}}$	$H_{n} (x)$

Each row is a geometry: a string, a sphere, a drum, a harmonic well. The special functions of mathematical physics are this one theorem applied to different domains and weights, which is worth knowing before you meet them one at a time.