Chapter 7: Sturm-Liouville Theory

A common theme runs through the process of solving most initial boundary value problems via the method of separation of variables. A solution is assumed to be a product of two of more functions each depending on a single variable. The functions which depend on the spatial variables (usually x, y, r, etc.) must satisfy the boundary conditions of the problem. This usually requires solving an ordinary differential equation with those associated boundary conditions, or as it has come to be called, a boundary value problem. See Eqs. (1.52)–(1.54) for a canonical example. The method of separation of variables usually introduces a constant which plays a role in the solution of the boundary value problem. Nontrivial solutions of the boundary value problem exist only for certain values of the constant. The values of the constant are called eigenvalues and the associated nontrivial solutions to the boundary value problem are called eigenfunctions. The boundary value problems encountered in the previous chapters on the heat, wave, and Laplace’s equation possess eigenfunctions forming orthogonal sets which, as a consequence of the results on Fourier series described in Chap. 3, allow a sufficiently smooth function, at least formally, to be represented as an infinite series whose terms are constant multiples of eigenfunctions. In many cases, this enabled the initial condition of the initial boundary value problem to be satisfied as well as the boundary conditions.