---
tag: 7WV6B2502K
kind: concept
name: Sturm–Liouville theory
msc: [34B24, 34L10, 34B09]
sources:
  - url: https://en.wikipedia.org/wiki/Sturm–Liouville_theory
same_as:
  wikidata: Q1154082
  wikipedia: Sturm–Liouville theory
---

## Summary

Sturm–Liouville theory studies the eigenvalue problems $-(p y')' + q y = \lambda w y$ on an interval, with boundary conditions at the ends.
For a regular problem with separated boundary conditions, the eigenvalues are real and simple and form a sequence $\lambda_1 < \lambda_2 < \cdots$ that tends to $+\infty$.
The $n$-th eigenfunction has exactly $n - 1$ zeros inside the interval, and the eigenfunctions form an orthogonal basis of the weighted space $L^2((a, b), w)$.
The theory explains Fourier series and many classical special functions, and it is the model case of the spectral theory of [[self-adjoint operators|0NEGN5CRSC]].

## Explanations

### L1 human

A vibrating string of varying thickness can only vibrate at certain frequencies.
Sturm–Liouville theory finds these frequencies as the [[eigenvalues|0BQAA2BK3Q]] of a second-order differential equation with conditions at the two ends of the string.
For a uniform string on $[0, \pi]$ fixed at both ends, the equation is $-y'' = \lambda y$ with $y(0) = y(\pi) = 0$.
The eigenvalues are $1, 4, 9, \ldots$ with the shapes $\sin x, \sin 2x, \sin 3x, \ldots$.
The $n$-th shape crosses zero $n - 1$ times between the ends, and the shapes are building blocks for any other shape.