In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form $${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y}$$ for given functions ${\displaystyle p(x)}$, ${\displaystyle q(x)}$ and ${\displaystyle w(x)}$, together with some boundary conditions at extreme values of ${\displaystyle x}$. The goals of a given Sturm–Liouville problem are:

Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions.

This theory is important in applied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a Sturm–Liouville problem.

Sturm–Liouville theory is named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), who developed the theory.

The main results in Sturm–Liouville theory apply to a Sturm–Liouville problem

on a finite interval ${\displaystyle [a,b]}$ that is "regular". The problem is said to be regular if:

The function ${\displaystyle w=w(x)}$, sometimes denoted ${\displaystyle r=r(x)}$, is called the weight or density function.

The goals of a Sturm–Liouville problem are: