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The Behavior of the Phase: The Oscillation Theorem

The answer is yes. Indeed, let be that solution to the Prüfer d.e. which satisfies the initial condition . We have one such solution for each . We can draw the graphs of these solutions for various values of . See Figure 3.13.

Note that if the function of the S-L equation is the constant function, then the slope

at every zero'' of will be a fixed constant, independent of . However, between a pair of successive zeroes'' the slope will be the larger, the larger is. Consequently, for large , the phase has an undulatory (i.e., wavelike'') behavior as passes through the successive zeroes of .

We now ask: How does

behave as a function of ? The answer to this question is important because it determines whether the other end point condition ( , ) can be satisfied.

The behavior of as a function of is summarized by the following three statements, which together comprise the

Oscillation Theorem:

The solution of the Prüfer d.e. satisfying the initial condition

1. is a continuous and strictly increasing function of ,
2. , i.e., is unbounded, and
for fixed in the interval . The line of reasoning leading to these conclusions consists of inferring properties about the -dependence of the function by making observations about the -dependence of its derivative , 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.

Next: Discrete Unbounded Sequence of Up: Phase Analysis of a Previous: The Boundary Value Problem   Contents   Index
Ulrich Gerlach 2010-12-09