The Behavior of the Phase: The Oscillation Theorem

The answer is yes. Indeed, let $\theta (x,\lambda )$ be that solution to the Prüfer d.e. which satisfies the initial condition $\theta (a,\lambda )= \gamma$ . We have one such solution for each $\lambda$ . We can draw the graphs of these solutions for various values of $\lambda$ . See Figure 3.13.

Figure 3.13: The graphs of the phase $\theta (x,\lambda )$ for the differential equation $u''+\lambda u=0$ with the indicated $\lambda$ -values. The initial phase value for all of them is $\gamma =\pi /2$ . Note that when the phase is an integral mulptiple of $\pi$ (dashed horizontal lines), all graphs have the slope $d\theta /dx=1$ .

Note that if the function $p(x)$ of the S-L equation is the constant function, then the slope

$$\frac{d\theta}{dx} =\frac{1}{p}~~\qquad~~(\textrm{when~}\theta =0,\pi ,2\pi , \dots )$$

at every ``zero'' of $u(x,\lambda )=r\sin\theta (x,\lambda )$ will be a fixed constant, independent of $\lambda$ . However, between a pair of successive ``zeroes'' the slope $\frac{d\theta}{dx}$ will be the larger, the larger $\lambda$ is. Consequently, for large $\lambda$ , the phase $\theta (x,\lambda )$ has an undulatory (i.e., ``wavelike'') behavior as $\theta (x,\lambda )$ passes through the successive zeroes of $u(x,\lambda)$ .

We now ask: How does

$$\theta (x,\lambda )$$

behave as a function of $\lambda$ ? The answer to this question is important because it determines whether the other end point condition ( $\theta (b,\lambda )=\delta +n\pi$ , $n=0,1,\dots$ ) can be satisfied.

The behavior of $\theta (x,\lambda )$ as a function of $\lambda$ is summarized by the following three statements, which together comprise the

Oscillation Theorem:

The solution $\theta (x,\lambda )$ of the Prüfer d.e. satisfying the initial condition

$$\theta (a,\lambda )=\gamma\,,~\quad~0\le\gamma <\pi~\quad~\forall~\lambda$$

1. is a continuous and strictly increasing function of $\lambda$ ,
2. $\lim\limits_{\lambda\to\infty}\theta (x,\lambda )=\infty$ , i.e., $\theta (x,\lambda )$ is unbounded, and
3. $\lim\limits_{\lambda\to -\infty} \theta (x,\lambda )=0$

for fixed $x$ in the interval $a<x\le b$ . The line of reasoning leading to these conclusions consists of inferring properties about the $\lambda$ -dependence of the function $\theta (x,\lambda )$ by making observations about the $\lambda$ -dependence of its derivative $$\frac{d\theta (x,\lambda)}{dx}$$ , which is given by the Prüfer d.e. (3.21). In spite of its straight forwardness, we shall not carry out this task at this time.