A Sturm-Liouville equation is a linear second-order ordinary differential equation of the form
![d/(dx)[p(x)(dy)/(dx)]+[lambdaw(x)-q(x)]y=0,](/images/equations/Sturm-LiouvilleEquation/NumberedEquation1.svg) |
(1)
|
where 
,
,
and 
are prescribed real functions, 
is a parameter, and 
is called the density or weighting function. The values
of 
for which a nonzero solution satisfies the prescribed
boundary conditions are the eigenvalues,
and the corresponding solutions 
are the eigenfunctions
(Arfken 1985).
A regular boundary value problem is posed on a finite closed interval ![[a,b]](/images/equations/Sturm-LiouvilleEquation/Inline8.svg)
, with 
having a continuous derivative,
and 
being continuous functions, and 
. Real homogeneous separated boundary
conditions independent of 
, such as Dirichlet
boundary conditions 
, give a self-adjoint
problem. Its eigenvalues are real and simple and form
a discrete set tending to 
(Teschl 2012, pp. 155-162). The eigenfunctions
can be chosen real and are orthogonal with
weight 
,
 |
(2)
|
for distinct eigenvalues 
.
For example, a uniform string fixed at both ends gives 
, 
, and the boundary value
problem
 |
(3)
|
with Dirichlet boundary conditions 
,
where 
.
Its eigenvalues and a corresponding choice of eigenfunctions
are
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
,
2, 3, ... (Byrne 2009, pp. 1-3). These eigenfunctions
give the standing-wave shapes obtained by separation
of variables in the one-dimensional
wave equation 
. For wave speed 
, the corresponding angular frequencies are 
.
Symbolic and numerical differential eigensystems can be computed in the Wolfram Language using DEigensystem and NDEigensystem, respectively. Variational approximations use the Rayleigh-Ritz variational technique. For example, VariationalBound in the Wolfram Language package VariationalMethods` and NVariationalBound give upper bounds on eigenvalues and optimized parameters for trial eigenfunctions.
Trott (2006, pp. 337-388) outlines the inverse Sturm-Liouville problem.
