Math 641-600 Final Exam - Hints

1. Most of the questions I've had about this problem have concerned errata, which are now fixed.

2. First of all, this theorem makes no assumptions about λ or ||K||. You only given that the inverse of I - λK exists. Thus you cannot directly use a Neumann series to find the inverse of I - λP_nK. A major step in doing that is to show that
   ||K - P_nK||_op → 0 as n → ∞.
   This makes use of the compactness of K. To see this, note that it is easy to show that ||I - P_n||_op = 1 for all n. Thus if we replace K by the identity I, the norm above would not go to zero. What you must make use of is that, for each fixed u ∈ H,
   ||u - P_nu|| → 0.
   

3. For questions about Green's functions, please see me.

4. Concerning 4(c), the graphs look better if you plot absolute values rather than squares of absolute values. (The squares of these quantities have more physical significance, though.) Also, rather than plotting the range of ω that you have available, namely 0 to 128π, try looking at the center of the plot, ω = 32π to 96π. This better illustrates the phenomenon mentioned in the last sentence.

Updated 12/11/06 (fjn).