Sturm-Liouville Theory: Eigenvalues, Eigenfunctions,

A Sturm-Liouville problem is the eigenvalue problem (p(x)y′)′ + (λw(x) − q(x))y = 0 on an interval [a, b], together with boundary conditions at the endpoints. It is the differential-equation analogue of the matrix eigenvalue problem Av = λv — except the "matrix" is a differential operator and the eigenvectors are functions, the eigenfunctions.

This theory is the hidden foundation of separation of variables. Every time a PDE solution is expanded in sines, cosines, Bessel functions, or Legendre polynomials, Sturm-Liouville theory is what guarantees those functions are orthogonal, complete, and come with the right eigenvalues. Without it, Fourier series and special-function expansions would be unjustified tricks.

Key Concepts

• Weight function w(x) > 0: defines the inner product; orthogonality of eigenfunctions holds with respect to this weight.
• Eigenvalues: form an increasing sequence λ1 < λ2 < … → ∞; only these discrete values admit solutions satisfying both boundary conditions.
• Orthogonality: ∫ab w(x)·ym(x)·yn(x) dx = 0 whenever m ≠ n.
• Rayleigh quotient: expresses each eigenvalue as a ratio of integrals, proving eigenvalues are real and giving estimates.
• Completeness: the eigenfunctions form a basis — any reasonable function on [a, b] expands in them (generalised Fourier series).

Formulas

• Sturm-Liouville form: (p(x)y′)′ + (λw(x) − q(x))y = 0, a < x < b, plus boundary conditions.
• Orthogonality: ∫ab w·ym·yn dx = 0 (m ≠ n).
• Rayleigh quotient: λ = (−[p·y′·y]ab + ∫ab(p(y′)2 + qy2)dx) / ∫ab w·y2 dx.
• Assumptions: p, w > 0, p′, q, w continuous. Limit: singular endpoints (Bessel at 0) need a limiting version of the theory.

Worked Example

Solve the Sturm-Liouville problem y″ + λy = 0 (so p = 1, w = 1, q = 0) with y(0) = 0, y(π) = 0. Trying y = sin(nx) gives y″ = −n2sin(nx), so λn = n2 with eigenfunctions yn = sin(nx), n = 1, 2, 3, … — eigenvalues 1, 4, 9, … marching to infinity as predicted. Check orthogonality for n = 1, m = 2: ∫0π sin x·sin 2x dx = ∫0π 2sin2x·cos x dx; substituting u = sin x (du = cos x dx) gives 2∫00 u2du = 0. Verify the Rayleigh quotient for y1 = sin x: the boundary term −[y′y]0π = 0, numerator ∫0πcos2x dx = π/2, denominator ∫0πsin2x dx = π/2, so λ = 1 — exactly λ1.

Common Misconceptions

• Any λ works. Only the discrete eigenvalues satisfy both boundary conditions; other λ values give only the trivial solution.
• Orthogonality needs no weight. The weight w(x) is essential — Bessel functions are orthogonal with weight x, not weight 1.
• Sturm-Liouville is only about sines and cosines. Bessel's and Legendre's equations are also Sturm-Liouville problems, which is why those functions are orthogonal too.

Real-World Applications

• Separation of variables for the heat and wave equations produces Sturm-Liouville eigenvalue problems; the expansions solve the PDE.
• Quantum mechanics: the time-independent Schrödinger equation is a Sturm-Liouville problem — its eigenvalues are the quantised energy levels.
• Vibrating strings and membranes: the modes are eigenfunctions, the audible frequencies come from √λn.

Sturm-Liouville theory studies (py′)′ + (λw − q)y = 0 with boundary conditions as an eigenvalue problem for differential operators. Its eigenvalues are real, discrete, and tend to infinity; its eigenfunctions are orthogonal with weight w and complete. This single framework justifies Fourier series, Bessel expansions, and Legendre expansions