Singular Boundary Value Problem:
Infinite Domain

All our observations of nature are specific and hence finite. In order to extend our grasp from the finite to those aspects nature termed ``infinite'', one starts with a one parameter family of (finite) systems. By letting this parameter become asymptotically large one can ask: are there any properties of the system that don't change as we let that parameter become arbitrarily large? An affirmative answer to this question yields a new perspective on the system. The constellation of properties subsumed under the existence of this mathematical limit is a new concept, the ``infinite system'' corresponding to the finite sytems giving rise to it.

The purpose of the method of the ``infinite system'' is to extend our grasp of nature from the direct perceptual level of awareness to the conceptual level which has no spatial and temporal bounds.

The archetypical system we shall study is the uniform string of length $\ell$ . We shall determine its response to a unit force, and after that let $\ell \to \infty$ . This is done in Section 4.10.4 on Page . We shall find the remarkable (but not necessarily unexpected) result that as $\ell \to \infty$ , the string's response becomes independent of the particular mixed Dirichlet-Neumann boundary conditions one may have imposed at $\ell$ .

It will turn out that when one lets $\ell$ become infinitely large, it is necessary to impose some other homogeneous boundary condition. The mixed D-N conditions are simply inappropriate for an (in the limit) infinite string. They get replaced by the so-called ``ingoing'' or ``outgoing'' wave conditions.

In order to become familiar with the key attributes of an infinite string consider again the differential equation

$$\frac{d^2 \phi}{dx^2} +\lambda \phi =0~, \quad -\infty <x<\infty$$

but without specifying any boundary conditions as yet. The general solution to this differential equation has the form

$$\begin{array}{rclrc} \phi_\lambda (x) &=& A~+~Bx & \textrm{f... ...i\lambda^{1/2} x} & \textrm{for}~~\lambda & \ne 0 \end{array}$$

The expression for $\phi _\lambda (x)$ is a $\lambda$ -parametrized family of solutions to a $\lambda$ -parametrized family of differential equations. The parameter may be, and in general is, complex. Consequently, $\phi _\lambda (x)$ should be viewed as a function of the complex variable $\lambda$ .

Figure 4.9: The $\lambda$ -parametrized family of functions $\phi _\lambda (x)$ as a map from the complex $\lambda$ -plane onto the complex $\phi$ -plane

For fixed $x$ this function maps the complex $\lambda$ -plane into the complex $\phi$ -plane

$$\begin{array}{rcl} \textrm{Complex}&~~~~~~\phi(x)~~~~~~&\tex... ... Ce^{i\sqrt{\lambda} x} ~+~De^{-i\sqrt{\lambda} x} \end{array}$$

The analytic properties of this function depend on the properties of the square root function $\lambda ^{1/2}$ , which has two analytic branches

$$\lambda^{1/2}=\alpha +i\beta=\left\{ \begin{array}{cl} \sqrt{\lambda}&\beta>0\\ -\sqrt{\lambda}&\beta<0 \end{array}\right.$$

and a branch cut which separates them. They all play a key role in determining the behaviour of the function $\phi _\lambda (x)$ . Thus a quick review is appropriate.