Manuscript Em02-E@1 · Em02-E-v1.tex · section · f3795050

The spatial inverse and its boundary condition

Clock slices and harmonic spatial labels › The spatial inverse and its boundary condition · lines 576–603

Em02-E@1#eq:Coulomb

Fix

(G_{Δ} F) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{F ( y )}{∣ x - y ∣} d^{3} y, F \in C_{c}^{\infty} (R^{3}) .

Then $Δ G_{Δ} F = F$ , and $G_{Δ} F \to 0$ at spatial infinity. For compact smooth $F$ , also $G_{Δ} Δ F = F$ . The sign in (eq:Coulomb) is fixed by $Δ (- 1/ (4 π ∣ x ∣)) = δ^{(3)}$ .

Seek harmonic rods on the clock slice in the form

Y_{λ}^{i} (s, x) = x^{i} + λ Y_{1}^{i} (s, x) + \frac{λ ^{2}}{2} Y_{2}^{i} (s, x), Δ_{q_{λ}} Y_{λ}^{i} = 0,

with $Y_{λ}^{i} - x^{i} \to 0$ at spatial infinity coefficientwise.

Displayed formulas in this passage

Em02-E@1#eq:Coulomb · line 579
$(G_{Δ} F) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{F ( y )}{∣ x - y ∣} d^{3} y, F \in C_{c}^{\infty} (R^{3}) .$
unlabeled · line 594
$Y_{λ}^{i} (s, x) = x^{i} + λ Y_{1}^{i} (s, x) + \frac{λ ^{2}}{2} Y_{2}^{i} (s, x), Δ_{q_{λ}} Y_{λ}^{i} = 0,$

The spatial inverse and its boundary condition

Displayed formulas in this passage

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