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

Lemma: Time-split expansion

Exact Friedrichs observables and analytical checks · lines 1103–1147 · proof included or linked

I-B-VP@3#lem:expansion

For $r = r^{'}$ , $Δ φ = Δ z = 0$ , and $Δ τ = s$ ,

H_{F, E} (s) = \frac{P _{β}}{48 π ^{2} r ^{2}} - \frac{C _{β} + 10 P _{β}}{2880 π ^{2} r ^{4}} s^{2} + O (s^{4}) .

Proof. The angular sum is

S_{β} (a) = \frac{cosh (( 1/2 - β ) a )}{sinh ( a /2 )} = \frac{2}{a} + B_{2} (β) a + \frac{B _{4} ( β )}{12} a^{3} + O (a^{5}),

where

B_{2} (β) = β^{2} - β + \frac{1}{6}, B_{4} (β) = β^{4} - 2 β^{3} + β^{2} - \frac{1}{30} .

Also

η = 2 arsinh \frac{s}{2 r}, η^{- 2} = \frac{r ^{2}}{s ^{2}} + \frac{1}{12} - \frac{s ^{2}}{240 r ^{2}} + O (s^{4}),

and

\frac{S _{β} ( η / α )}{α sinh η} = \frac{2}{η ^{2}} + \frac{B _{2}}{α ^{2}} - \frac{1}{3} + η^{2} [\frac{B _{4}}{12 α ^{4}} - \frac{B _{2}}{6 α ^{2}} + \frac{7}{180}] + O (η^{4}) .

Multiply by $(8 π^{2} r^{2})^{- 1}$ , subtract $(4 π^{2} s^{2})^{- 1}$ , and use $P_{β} = 6 B_{2} / α^{2} - 1$ , $C_{β} = - 30 B_{4} / α^{4} - 1$ . Theorem thm:analytic identifies these with the unique regular coincidence limits.

Displayed formulas in this passage

unlabeled · line 1107
$H_{F, E} (s) = \frac{P _{β}}{48 π ^{2} r ^{2}} - \frac{C _{β} + 10 P _{β}}{2880 π ^{2} r ^{4}} s^{2} + O (s^{4}) .$
unlabeled · line 1117
$S_{β} (a) = \frac{cosh (( 1/2 - β ) a )}{sinh ( a /2 )} = \frac{2}{a} + B_{2} (β) a + \frac{B _{4} ( β )}{12} a^{3} + O (a^{5}),$
unlabeled · line 1123
$B_{2} (β) = β^{2} - β + \frac{1}{6}, B_{4} (β) = β^{4} - 2 β^{3} + β^{2} - \frac{1}{30} .$
unlabeled · line 1128
$η = 2 arsinh \frac{s}{2 r}, η^{- 2} = \frac{r ^{2}}{s ^{2}} + \frac{1}{12} - \frac{s ^{2}}{240 r ^{2}} + O (s^{4}),$
unlabeled · line 1134
$\frac{S _{β} ( η / α )}{α sinh η} = \frac{2}{η ^{2}} + \frac{B _{2}}{α ^{2}} - \frac{1}{3} + η^{2} [\frac{B _{4}}{12 α ^{4}} - \frac{B _{2}}{6 α ^{2}} + \frac{7}{180}] + O (η^{4}) .$

Lemma: Time-split expansion

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