In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form
d d x [ p ( x ) d y d x ] + q ( x ) y = − λ w ( x ) y {\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 p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)} and w ( x ) {\displaystyle w(x)} , together with some boundary conditions at extreme values of x {\displaystyle x} . The goals of a given Sturm–Liouville problem are:
To find the λ {\displaystyle \lambda } for which there exists a non-trivial solution to the problem. Such values λ {\displaystyle \lambda } are called the eigenvalues of the problem. For each eigenvalue λ {\displaystyle \lambda } , to find the corresponding solution y = y ( x ) {\displaystyle y=y(x)} of the problem. Such functions y {\displaystyle y} are called the eigenfunctions associated to each λ {\displaystyle \lambda } . 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.
Main results The main results in Sturm–Liouville theory apply to a Sturm–Liouville problem
on a finite interval [ a , b ] {\displaystyle [a,b]} that is "regular". The problem is said to be regular if:
the coefficient functions p , q , w {\displaystyle p,q,w} and the derivative p ′ {\displaystyle p'} are all continuous on [ a , b ] {\displaystyle [a,b]} ;
p ( x ) > 0 {\displaystyle p(x)>0} and w ( x ) > 0 {\displaystyle w(x)>0} for all x ∈ [ a , b ] {\displaystyle x\in [a,b]} ; the problem has separated boundary conditions of the form
The function w = w ( x ) {\displaystyle w=w(x)} , sometimes denoted r = r ( x ) {\displaystyle r=r(x)} , is called the weight or density function. The goals of a Sturm–Liouville problem are:
to find the eigenvalues: those λ for which there exists a non-trivial solution; for each eigenvalue λ, to find the corresponding eigenfunction y = y ( x ) {\displaystyle y=y(x)} . For a regular Sturm–Liouville problem, a function y = y ( x ) {\displaystyle y=y(x)} is called a solution if it is continuously differentiable and satisfies the equation (1) at every x ∈ ( a , b ) {\displaystyle x\in (a,b)} . In the case of more general p , q , w {\displaystyle p,q,w} , the solutions must be understood in a weak sense. The terms eigenvalue and eigenvector are used because the solutions correspond to the eigenvalues and eigenfunctions of a Hermitian differential operator in an appropriate Hilbert space of functions with inner product defined using the weight function. Sturm–Liouville theory studies the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in the function space. The main result of Sturm–Liouville theory states that, for any regular Sturm–Liouville problem:
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