The Boundary Conditions

The two D-N boundary conditions are

$$\alpha u(a)+\alpha 'u'(a)=0~~\quad~~\textrm{and}~~\quad~~\beta u(b)+\beta ' u'(b)=0$$

at the two endpoints $x=a$ and $x=b$ . We know that the phase $\theta (x,\lambda )$ satisfies the family of $\lambda$ -parametrized $1^{\textrm{st}}$ order o.d.e.'s

$$\frac{d\theta}{dx} = (\lambda\rho (x)-q(x))\sin^2\theta +\frac{1}{p(x)}\cos^2 \theta ~~,$$

where $\rho$ , $q$ , and $p$ are given by the S-L equation

$$\left[\frac{d}{dx} p(x)\frac{d}{dx} +\lambda\rho (x)-q(x)\right] u=0\,.$$

We must now determine what conditions the two homogeneous D-N boundary conditions impose on the phase $\theta (x)$ . The transformation of the D-N conditions into equivalent conditions on the phase is done with the help of the Prüfer relation

$$\tan\theta =\frac{u}{pu'}\,.$$

This determines two phase angles. At the left endpoint $x=a$ , let the initial phase be $\theta (a,\lambda )= \gamma$ . This phase $\gamma$ is uniquely determined by the two requirements

$$\tan\gamma \equiv \frac{u(a)}{u'(a) p(a)}=-\frac{\alpha '}{\alpha p(a)}~~ \textrm{and}~~\boxed{0\le\gamma <\pi}~~,$$

if $\alpha\not= 0$ , and by

$$\gamma =\frac{\pi}{2} ~~\textrm{if}~~\quad ~\alpha =0~~.$$

(It is clear that $\gamma =\frac{\pi}{2}$ expresses the case of pure Neumann condition at $x=a.$ ) Thus the D-N boundary condition at $x=a$ has been expressed in terms of a single quantity, the initial phase. This initial phase is required to be the same for all $\lambda$ .

At $x=b$ we introduce the final phase angle $\delta$ . It is determined by the two requirements

$$\tan\delta = -\frac{\beta'}{\beta p(b)}~~\textrm{and}~~\boxed{0<\delta\le\pi}$$

if $\beta \ne 0$ , and by

$$\delta =\frac{\pi}{2}~~\quad\textrm{if}~~\quad~\beta =0\,.$$

Lecture 24

Having reformulated the two D-N conditions in terms of the two angles $\gamma$ and $\delta$ , we are ready to restate the S-L problem in terms of the phase function $\theta$ . This restatement is very simple.