Sturm–Liouville Theory — Mathematics Knowledge System

Overview

Sturm–Liouville theory provides a unified framework for the eigenvalue problems that arise from separating variables in PDEs. A Sturm–Liouville problem consists of a self-adjoint differential operator L[y] = −(p(x)y′)′ + q(x)y on an interval [a,b] with appropriate boundary conditions, and an eigenvalue equation L[y] = λw(x)y (where w > 0 is a weight function). The theory guarantees: real eigenvalues, an infinite sequence λ₁ < λ₂ < … → +∞, orthogonal eigenfunctions with respect to the weight w, and completeness (every square-integrable function on [a,b] can be expanded in the eigenfunctions). Fourier series, Bessel series, and Legendre series are all special cases.

Intuition

Sturm–Liouville theory is the ODE analogue of the spectral theorem for symmetric matrices. Just as a real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors (so any vector can be expanded in them), a Sturm–Liouville operator has real eigenvalues and a complete orthogonal basis of eigenfunctions. The Fourier series is the simplest example: the operator −d²/dx² on [0,L] with Dirichlet conditions has eigenfunctions sin(nπx/L) and eigenvalues (nπ/L)². All classical series expansions in mathematical physics are instances of this general principle.

Formal Definition

Definition

A regular Sturm–Liouville problem on [a,b] consists of the equation and separated boundary conditions:

L[y] = -(p(x)y')' + q(x)y = \lambda w(x) y,\quad x \in [a,b]

Sturm–Liouville equation

\alpha_1 y(a) + \alpha_2 y'(a) = 0,\quad \beta_1 y(b) + \beta_2 y'(b) = 0

Separated boundary conditions

\int_a^b y_m(x)\, y_n(x)\, w(x)\,dx = 0 \quad (m \neq n)

Orthogonality of eigenfunctions

f(x) = \sum_{n=1}^{\infty} c_n y_n(x),\quad c_n = \frac{\int_a^b f(x)y_n(x)w(x)\,dx}{\|y_n\|_w^2}

Eigenfunction expansion

Worked Examples

This is the Sturm–Liouville problem with p=1, q=0, w=1, a=0, b=π.
$-y'' = \lambda y,\quad y(0)=y(\pi)=0$
For λ>0 write λ=k². General solution: y = A sin(kx) + B cos(kx). y(0)=B=0, so y=A sin(kx).
$y(0) = B = 0 \Rightarrow y = A\sin(kx)$
y(π)=0: A sin(kπ)=0. For nontrivial solutions: kπ = nπ, so k=n for n=1,2,3,…
$k = n \in \mathbb{Z}^+$
Eigenvalues: λₙ = n². Eigenfunctions: yₙ(x) = sin(nx), forming the Fourier sine basis.
$\lambda_n = n^2,\quad y_n(x) = \sin(nx)$

Answer: λₙ = n², yₙ(x) = sin(nx) for n = 1, 2, 3, …

Practice Problems

Difficulty 7/10

Find the eigenvalues and eigenfunctions of −y″ = λy on [0,1] with y′(0)=y′(1)=0 (Neumann BCs).

Difficulty 8/10

State the completeness theorem for Sturm–Liouville eigenfunctions and explain its significance.

Difficulty 9/10

Show that eigenfunctions of a regular SL problem corresponding to distinct eigenvalues are orthogonal with respect to the weight w.

Common Mistakes

Common Mistake

Eigenvalues of a SL problem can be complex.

The self-adjointness of the SL operator guarantees all eigenvalues are real, and for regular problems they form an increasing sequence tending to +∞.

Common Mistake

Different boundary conditions give the same eigenfunctions.

The boundary conditions fundamentally affect which eigenfunctions arise: Dirichlet gives sines, Neumann gives cosines, periodic gives both, etc.

Quiz

What property of the Sturm–Liouville operator ensures real eigenvalues?

Positivity of p(x)Self-adjointness (symmetry) in L²([a,b]; w dx)Compactness of the resolventPositivity of w(x)

The Fourier sine series is the eigenfunction expansion for which SL problem?

−y″=λy on [0,L] with y(0)=y(L)=0−y″=λy on [0,L] with y′(0)=y′(L)=0−(xy′)′=λxy on [0,1]−y″+y=λy on ℝ

Summary

−(p y′)′ + q y = λ w y with separated BCs is the Sturm–Liouville problem.
Eigenvalues are real and form an increasing sequence λ₁ < λ₂ < … → +∞.
Eigenfunctions for distinct eigenvalues are orthogonal in L²([a,b]; w dx).
The eigenfunctions form a complete basis: any function in L² has a generalized Fourier expansion.
Fourier, Bessel, and Legendre series are all special cases of SL eigenfunction expansions.

References

WebsiteWikipedia — Sturm–Liouville theory