Old work from Appendix A
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Short unused note dated 3.26.05 from the "stuff not used" folder of Phil's Spectral Theory Book. It shows that the central peak of the kernel delta5(k,N) has integral 1 as N goes to infinity, using the small-k approximation sin(k/2)=k/2 and the Sine Integral Si(x), whose limit is pi/2. It concludes the kernel tends to a sum of delta functions at k = 2 pi m. Equations are partly lost to conversion errors.
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Old work from Appendix A PhL 3.26.05
Consider for example the central peak. We want to show that
!Syntax Error, Idk { limN→∞ } = 1 for arbitrarily small a (A.18)
We have already argued that {...} will have zero integral over any region away from k = 0, so the entire integral must come from the region immediately close to k = 0. In this neighborhood, we can make the approximation sin(k/2) = k/2, so our task is then to show that
!Syntax Error, Idk { limN→∞ } = 1 for arbitrarily small a (A.19)
But this expression is not too hard to evaluate. We find
!Syntax Error, Idk δ5(k,N) = !Syntax Error, Idk { limN→∞ }
= limN→∞ ( !Syntax Error, Idk )
= limN→∞ ( Si[(N+1/2)a] )
= limN→∞ Si[(N+1/2)a] = = 1. (A.20)
The function Si(x) is the Sine Integral function, and it has the property limx→∞ Si(x) = π/2 . Thus, for an arbitrarily small a > 0, the limit is as shown above. Here is the integral done in Maple,
The same analysis may be applied for any of the delta peaks. Thus we have demonstrated that
limN→∞ δ5(k,N) = !Syntax Error, Iδ(k - 2πm) (A.21)