Green's Function as the Fountainhead of the Eigenvalues and Eigenvectors of a System

The spectral representation of $G$ is a second way of writing the Green's function. The first way was Eq.(4.19) on page  in terms of the two independent solutions (``elements of the null space'') of the homogeneous Eq.(4.24) on page  with $f=0$ .

This raises an important question: Are the null space representation and the spectral representation really one and the same function? The answer is, of course, ``yes'' because of the uniqueness of the Green's function for a given problem. This fact was the subject of Green's function uniqueness theorem on page .

We shall now take advantage of this uniqueness in order to obtain from the null space representation of the Green's function the complete set of orthonormal eigenfunctions of the linear system. The beauty of this method is that one directly obtains the eigenvalues and these eigenfunctions without having to evaluate any normalization integrals. Thus the Green's function of a system lives up to its reputation of being able to give everything one wants to know about the internal workings of a linear system but never dared to ask.

However, the story does not end there. The Green's function gives one a quick method for checking the completeness of a set of eigenfunctions, i.e. whether they span the whole vector space of functions satisfying the same homogeneous boundary conditions. This completeness was already validated in Section 3.7 (page ) using Rayleigh's quotient as the starting point. But with the Green's function at hand, the completeness property can be easily validated by merely evaluating an integral.

Figure 4.7: The integration path in the complex $\lambda$ -plane is to be large enough to enclose all the eigenvalues of the Sturm-Liouville problem, the poles of the Green's function. That integration path is deformed into a union of small circles, each one enclosing a single eigenvalue.