Sturm-Liouville Systems

One of the most important and best understood eigenvalue problems in linear algebra is

$$(A-\lambda B)u=0\,,$$

where $A$ is a symmetric matrix and $B$ is a symmetric positive definite matrix. For this problem we know that

1. its eigenvalues form a finite sequence of real numbers
2. the eigenvectors form a $B$ -orthogonal basis for the vector space; in other words,
   $$U^T_i BU_j=\delta_{ij}\,.$$

A Sturm-Liouville system extends this eigenvalue problem to the framework of $2^{\textrm{nd}}$ order linear ordinary differential equations (o.d.e.'s) where the vector space is infinite dimensional, as we shall see.