Spectral Representation

Consider the spectral representation of the Green's function $G_\lambda(x;\xi)$ . Viewed as a function of the variable $\lambda$ , this function has poles in the complex $\lambda$ -plane. These poles are the eigenvalues of the Sturm-Liouville problem. If the problem is self-adjoint, these poles lie along the real axis (Theorem 2 on page ). However, in general they may lie anywhere in the complex plane.

Suppose we consider the contour integral

$$\frac{1}{2\pi i} \oint G_\lambda (x;\xi)~d\lambda$$

around a closed integration path which is large enough to enclose all the poles of the Green's function. According to the Cauchy-Goursat theorem, if one deforms the integration contour of this integral from the large circle in Figure 4.7 into the union of the little circles around the poles $\lambda_0,\lambda_1,\lambda_2, \cdots$ of the integrand, then the value of the integral will not change; in other words,

$$\frac{1}{2\pi i} \oint_C G_\lambda (x;\xi)~d\lambda = \frac{1}{2\pi i} \oint_{{\bigcup C_n \atop n~~~}} G_\lambda (x;\xi)~d\lambda$$

$$= \sum_{n=0}^\infty \frac{1}{2\pi i} \oint_{C_n} G_\lambda (x;\xi)~d\lambda$$ (427)

Each term on the right hand side equals the residue of $G_\lambda$ at its respective pole $\lambda_n$ . According to Eq.(4.26) this residue is

$$\begin{array}[t]{c} \textrm{Res}\\ \lambda =\lambda _n \end{array}G_\lambda (x;\xi) = \lim_{\lambda \to \lambda _n} (\lambda -\lambda _n) G_\lambda (x;\xi)$$

$$~ = -u_n(x)\overline u_n(\xi)$$

Thus

$$\sum_{n=0}^\infty \frac{1}{2\pi i} \oint_{C_n} G_\lambda (x;\xi)~d\lambda= -\sum_{n=0}^\infty u_n(x)\overline u_n(\xi)$$ (428)

Consequently, the contour integral, Eq.(4.27), is

$$\boxed{ \frac{1}{2\pi i} \oint_C G_\lambda (x;\xi)~d\lambda=-\sum_{n=0}^\infty u_n(x)\overline u_n(\xi)}~~,$$ (429)

This is a remarkable relation. It says that if one somehow can determine the $\lambda$ -parametrized Green's function for the problem

$$\left( \frac{d}{dx} p(x) \frac{d}{dx}-q(x) + \lambda \rho(x) \right) G_\lambda (x;\xi) = -\delta (x-\xi)$$

$$B_1(G_\lambda) =$$ 0

$$B_2(G_\lambda) = 0~~,$$

then this function yields the corresponding complete set of orthonormalized eigenfunctions of the Sturm-Liouville operator. This can be done whenever one can find a closed expression for $G_\lambda(x;\xi)$ . The example below illustrates this.