Manuscript X-A-conv@1 · X-A-conv.tex · proposition · c76fffa3

Proposition: Coordinate coefficients and uniqueness

Cauchy operators and the regulated coordinate problem › The actual regulated problem · lines 921–964 · proof included or linked

X-A-conv@1#prop:coordcoeff, X-A-conv@1#eq:w, X-A-conv@1#eq:v

The unique coefficients satisfying (eq:regcoord)–(eq:coorddata) are

w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} (H), = E_{0} C^{μ} (J) - E_{0} [\partial_{α} a_{2} (H)^{α μ} + \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ} [h])] .

Both coefficients and their first time derivatives vanish on $Σ$ .

Proof. The auxiliary density expansion is

A (q_{g, ε}) = η + κ a_{1} (H) + κ^{2} (a_{1} (J) + a_{2} (H)) (mod κ^{3}) .

Substitution in (eq:regcoord) gives

□_{η} w^{μ} = C^{μ} (H),

□_{η} v^{μ} = C^{μ} (J) - \partial_{α} a_{2} (H)^{α μ} - \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ}) .

The sources are smooth. Lemma lem:Cauchy gives the formulas and successive uniqueness.

Displayed formulas in this passage

X-A-conv@1#eq:w · line 925
$w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} (H), = E_{0} C^{μ} (J) - E_{0} [\partial_{α} a_{2} (H)^{α μ} + \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ} [h])] .$
X-A-conv@1#eq:v · line 925
$w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} (H), = E_{0} C^{μ} (J) - E_{0} [\partial_{α} a_{2} (H)^{α μ} + \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ} [h])] .$
unlabeled · line 945
$A (q_{g, ε}) = η + κ a_{1} (H) + κ^{2} (a_{1} (J) + a_{2} (H)) (mod κ^{3}) .$
unlabeled · line 952
$□_{η} w^{μ} = C^{μ} (H),$
unlabeled · line 955
$□_{η} v^{μ} = C^{μ} (J) - \partial_{α} a_{2} (H)^{α μ} - \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ}) .$