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Concept# Sturm–Liouville theory

Summary

In mathematics and its applications, classical Sturm–Liouville theory (developed by Joseph Liouville and Jacques Charles François Sturm) is the theory of real second-order linear ordinary differential equations of the form:
for given coefficient functions , , and , an unknown function of the free variable , and an unknown constant . All homogeneous (i.e. with the right-hand side equal to zero) second-order linear ordinary differential equations can be reduced to this form. In addition, the solution y is typically required to satisfy some boundary conditions at extreme values of x. Each such equation () together with its boundary conditions constitutes a Sturm–Liouville problem.
In the simplest case where all coefficients are continuous on the finite closed interval and p has continuous derivative, a function y = y(x) is called a solution if it is continuously differentiable and satisfies the equation () at every . In the case of more general p(x), q(x), w(x), the solutions must be understood in a weak sense.
The function w(x), sometimes denoted r(x), is called the weight or density function.
The value of λ is not specified in the equation: finding the λ for which there exists a non-trivial solution is part of the given Sturm–Liouville problem. Such values of λ, when they exist, are called the eigenvalues of the problem, and the corresponding solutions are the eigenfunctions associated to each λ. This terminology is 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.
This theory is important in applied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations.

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