Sturm-Liouville Theory

The chapter starts by considering ODEs subject to boundary conditions, and then considers the conditions under which the operator defining the ODE, together with the boundary conditions is Hermitian. This analysis, Sturm-Liouville theory, includes techniques for making an operator self-adjoint by multiplying its ODE by a weight factor and making an appropriate definition of the scalar product. The use of specific properties of ODEs to aid in the solution of boundary-value problems is illustrated for the Legendre and Hermite operators, and a two-region problem is used to illustrate the effect of matching conditions. The variation method (common in quantum mechanics) is described and illustrated.