Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · proposition · f4bd2b11

Proposition: Global regularized covariance

An exact summed kernel and its boundary values › Summation before continuation · lines 502–553 · proof included or linked

I-B-VP-2@1#prop:closed, I-B-VP-2@1#eq:Wclosed

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For $ε > 0$ ,

W_{β, ε} (X, X^{'}) = \frac{S _{β} ( η _{ε} , δ )}{8 π ^{2} α r r ^{'} sinh η _{ε}} = [\frac{e ^{- (ε + i T) H_{β}}}{2 H _{β}}] (x, x^{'}) .

Its distributional boundary value as $ε ↓ 0$ is $W_{β}$ .

Proof. For a real Euclidean separation $a_{0} > 0$ , the spectral identity

\frac{e ^{- a_{0} B}}{2 B} = \int_{0}^{\infty} (4 π s)^{- 1/2} e^{- a_{0}^{2} / (4 s)} e^{- s B} d s

and (eq:heat) give

G_{E, β} = \frac{S _{β} ( η , δ )}{8 π ^{2} α r r ^{'} sinh η}, cosh η = \frac{r ^{2} + r ^{'2} + Z ^{2} + a _{0} ^{2}}{2 r r ^{'}} .

The heat integral is justified at large $s$ by Lemma lem:IR. The change of variable $ζ = r r^{'} / (2 s)$ uses

\int_{0}^{\infty} e^{- ζ c o s h η} I_{ν} (ζ) d ζ = \frac{e ^{- ν η}}{sinh η}

Watson. This is also the Euclidean mode kernel of I-B-VP and I-B-VP-0 CheungIBVP,CheungIBVPZero.

Both sides continue holomorphically in $a_{0}$ to $Re a_{0} > 0$ . The denominators in (eq:Sgeom) are nonzero there under (eq:Qeta), since $Re η > 0$ . Set $a_{0} = ε + i T$ . Spectral dominated convergence on the compact-test form of Lemma lem:IR proves the boundary assertion.

Displayed formulas in this passage

I-B-VP-2@1#eq:Wclosed · line 505
$W_{β, ε} (X, X^{'}) = \frac{S _{β} ( η _{ε} , δ )}{8 π ^{2} α r r ^{'} sinh η _{ε}} = [\frac{e ^{- (ε + i T) H_{β}}}{2 H _{β}}] (x, x^{'}) .$
unlabeled · line 522
$\frac{e ^{- a_{0} B}}{2 B} = \int_{0}^{\infty} (4 π s)^{- 1/2} e^{- a_{0}^{2} / (4 s)} e^{- s B} d s$
unlabeled · line 528
$G_{E, β} = \frac{S _{β} ( η , δ )}{8 π ^{2} α r r ^{'} sinh η}, cosh η = \frac{r ^{2} + r ^{'2} + Z ^{2} + a _{0} ^{2}}{2 r r ^{'}} .$
unlabeled · line 539
$\int_{0}^{\infty} e^{- ζ c o s h η} I_{ν} (ζ) d ζ = \frac{e ^{- ν η}}{sinh η}$

Proposition: Global regularized covariance

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