Definition:Sturm-Liouville Theory/Historical Note

Historical Note on Sturm-Liouville Theory
Under normal assumptions on the coefficient functions $\map p x$, $\map q x$, and $\map w x$ above, they induce a Hermitian differential operator in some function space defined by boundary conditions.

The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in a suitable function space became known as Sturm-Liouville theory. This theory is important in applied mathematics, where S-L problems occur very commonly, particularly when dealing with linear partial differential equations that are separable.