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## Discrete Unbounded Sequence of Eigenvalues

With this Oscillation theorem'' tells us that is a function which increases without limit as . Consequently, as increases from , there will be a first value, say , for which the second boundary condition (the one at ), i.e.,

is satisfied. Moreover, as increases beyond , increases monotonically beyond until it reaches the value . This happens at a specific value of , say , which is larger than ,

Continuing in this fashion, one finds that, regardless of how big an integer one picks, the equation

always has a solution for , which we shall call . This yields an infinite discrete sequence of 's which is monotonically increasing

This sequence has no upper bound. Why? For any large consider . This number lies between some pair of points, say,

The Oscillation Theorem says that has the property of being a monotonic function of whose range is the whole positive real line. The latter property guarantees that each of the two equations,

and

has a solution. The former property guarantees that each of these two solutions is unique. Call them and . The former property also guarantees that

Since can be as large as we please, the sequence of eigenvalues,

has no upper bound.

Corresponding to this sequence, there is the set of eigenfunctions

Each of these functions oscillates as a function of . How many times does each pass through zero in the open interval ? Reference to Figure 3.13 shows that has precisely zeroes inside ; zeroes at the endpoints, if any, do not count. Indeed, it must have at least zeroes because the graph of with held fixed as in Figure 3.13, must cross at least multiples of (dotted horizontal lines in Figure 3.13 and 3.11). On the other hand, the function cannot have more than zeroes because the graph of phase can cross each multiple of no more than once. This fact is guaranteed by Eq.(3.25) on page .

To summarize, we have the following

Theorem: Any regular S-L problem has an infinite number of solutions which belong to the real eigenvalues

Furthermore, each eigenfunction
1. has exactly zeroes in the interval ,
2. is unique up to a constant multiplicative factor.

Lecture 25

Next: Completeness of the Set Up: Phase Analysis of a Previous: The Behavior of the   Contents   Index
Ulrich Gerlach 2010-12-09