Qualitative Results

The phase is a very direct way of deducing a number of important properties of any solution to a general second order linear o.d.e. We shall do this by making a number of simple observations.

(i) The zeroes of a solution $u(x)$ to Eq. 3.18 occur where the Prüfer phase $\theta$ has the values

$$0,\pm\pi ,\pm 2\pi ,\dots$$

Figure 3.9: The function $u(x)$ has its zeroes whenever the phase $\theta (x)$ is an integral multiple of $\pi$ .

(ii) At these points, where $\sin\theta (x)=0$ , $\theta (x)$ is an increasing function of $x$ . Indeed, $\sin\theta =0$ implies $\cos^2\theta =1$ . Consequently, the Prüfer equation yields

$$\frac{d\theta}{dx} = \frac{1}{P}>0\,,~~\qquad~~\qquad~~(\textrm{whenever}~ \theta =0,\pm\pi ,\dots )$$ (325)

because $P(x)>0$ by assumption. The positiveness of this rate of change implies that in the Poincaré phase plane, which is spanned by $Pu'$ and $u$ , the curve $(P(x)u'(x),u(x))$ crosses the horizontal $Pu'$ -axis ( $\theta =n\pi$ ) only in the counter clockwise sense as illustrated in Figure 3.7. In other words, the phase of the curve always goes forward, never backward (Figure 3.10) when it crosses the horizontal.

Figure 3.10: The phase can only advance, never retreat across the horizontal axis.

The following conclusions follow easily.

(iii) The zeroes of $u(x)$ are isolated, i.e. they are separated by a finite amount from each other. Why is this statement true? Consider two successive zeroes of $u(x)$ . Call them $x_n$ and $x_{n+1}$ . At these points the phase has the values $\theta (x_n)=n\pi$ and $\theta (x_{n+1}) =(n+1)\pi$ . The phase equation

$$\frac{d\theta}{dx} = Q(x)\sin^2\theta +\frac{1}{P(x)}\cos^2\theta$$

implies that the slopes $\frac{d\theta}{dx}\Bigg\vert _{x_n} = \frac{1}{P(x_n)}$ and $\frac{d\theta}{dx}\Bigg\vert _{x_{n+1}} = \frac{1}{P(x_{n+1})}$ both be positive. Reference to the $\theta -x$ plane, say Figure 3.11, reveals that these two inequalities simply prevent $x_n$ and $x_{n+1}$ from being infinitesimally close together. The function $\theta (x)$ would become multivalued if they were.

Figure 3.11: The zeroes of $u(x)$ , i.e. the integral $\pi$ values of $\theta$ must have a finite $x$ -separation, otherwise $\theta (x)$ becomes multi-valued.

(iv) If $Q>0$ , then $u(x)$ has exactly one maximum (or minimum) between two successive zeroes of a given solution. Thus a sequence of maxima and minima above (or below) the $x$ -axis is impossible if $Q(x)>0$ .

The reason for this impossibility is this:

1. At a maximum (or a minimum) of $u$ one has
   $$0=Pu'=r\cos\theta~~~\Leftrightarrow~~~\cos\theta =0\,,~~\sin^2\theta =1~~.$$
   A maximum (or minimum) of $u$ is located at $\theta =(n+\frac{1}{2})\pi$ .
2. Prüfer's equation (3.21) implies, therefore,

   $$\frac{d\theta}{dx} = Q\sin^2\theta +0=Q>0~~\textrm{(at~a~MAX~or~a~MIN)}$$ (326)

   
   
   at these points. Consequently, $\theta (x)$ can cross the line $\theta =(n+\frac{1}{2})\pi$ only once, which means $u$ has a maximum (or minimum) only once. If it crossed it a second time, as in Figure 3.12, the slope would have to be negative at the second crossing point, thus violating the inequality 3.26.

If $Q<0$ , then, of course, all bets are off!

Figure 3.12: The minimum of $u(x)$ in this figure is forbidden because the corresponding slope $d\theta /dx$ at that point would have to be negative, in violation of inequality 3.26.