Manuscript I-B-VP@3 · I-B-VP-v3.tex · corollary · 93c04eb4

Corollary: Untwisted diffraction identity

Exact Friedrichs observables and analytical checks › Diffraction identity and quotient sums · lines 1284–1303 · proof included or linked

I-B-VP@3#cor:diffraction

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If $1/ (2 α) \in / N$ , then

0 < ∣ k ∣ < 1/ (2 α) \sum csc^{2} (π α k) - \frac{1}{π α} \int_{0}^{\infty} \frac{sin ( π / α ) d u}{[ cosh ( u / α ) - cos ( π / α )] cosh ^{2} ( u /2 )} = \frac{α ^{- 2} - 1}{3} .

For $α > 1/2$ , the image sum is empty, and on a sufficiently small diagonal neighbourhood the regular remainder is wholly represented by diffraction.

Proof. At coincidence $R_{k}^{2} = 4 r^{2} sin^{2} (π α k)$ and $ρ (u)^{2} = 4 r^{2} cosh^{2} (u /2)$ . Multiply the subtracted contour representation by $16 π^{2} r^{2}$ .

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$0 < ∣ k ∣ < 1/ (2 α) \sum csc^{2} (π α k) - \frac{1}{π α} \int_{0}^{\infty} \frac{sin ( π / α ) d u}{[ cosh ( u / α ) - cos ( π / α )] cosh ^{2} ( u /2 )} = \frac{α ^{- 2} - 1}{3} .$

Corollary: Untwisted diffraction identity

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