Definition:Sturm-Liouville Equation

Definition

A classical Sturm-Liouville equation is a real second order ordinary linear differential equation of the form:

$(1): \quad \displaystyle - \frac {\mathrm d} {\mathrm d x} \left({p \left({x}\right) \frac {\mathrm d y} {\mathrm d x}}\right) + q \left({x}\right) y = \lambda w \left({x}\right) y$

where $y$ is a function of the free variable $x$.

The functions $p \left({x}\right)$, $q \left({x}\right)$ and $w \left({x}\right)$ are specified.

In the simplest cases they are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

$(1a): \quad p \left({x}\right) > 0$ has a continuous derivative
$(1b): \quad w \left({x}\right) > 0$
$(1c): \quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$.

Weight Function

The function $w \left({x}\right)$, which is sometimes called $r \left({x}\right)$, is called the weight function or density function.

Eigenvalues

The value of $\lambda$ is not specified in the equation.

Finding the values of $\lambda$ for which there exists a non-trivial solution of $(1)$ satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).

Such values of $\lambda$ when they exist are called the eigenvalues of the boundary value problem defined by $(1)$ and the prescribed set of boundary conditions.

The corresponding solutions (for such a $\lambda$) are the eigenfunctions of this problem.

Source of Name

This entry was named for Jacques Charles François Sturm and Joseph Liouville.