Manuscript VII-A@1 · VII-A.tex · lemma · 11da3fe6

Lemma: The left-inverse identity with its data condition

Cauchy problems and the admitted gauge domain › Spacetime and free propagation · lines 337–363 · proof included or linked

VII-A@1#lem:leftinverse, VII-A@1#eq:leftinverse, VII-A@1#eq:generalleft

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If $ψ \in C^{\infty} (M_{+})$ satisfies

ψ ∣_{Σ_{0}} = 0, \partial_{t} ψ ∣_{Σ_{0}} = 0,

then

E_{m} P_{m} ψ = ψ .

More generally,

E_{m} P_{m} ψ = ψ - H_{m} [ψ],

where $H_{m} [ψ]$ is the homogeneous solution with the same Cauchy data as $ψ$ .

Proof. The difference $ψ - E_{m} P_{m} ψ$ solves the homogeneous equation and has the same Cauchy data as $ψ$ . Uniqueness gives (eq:generalleft); the zero-data case gives (eq:leftinverse).

Displayed formulas in this passage

VII-A@1#eq:leftinverse · line 345
$E_{m} P_{m} ψ = ψ .$
VII-A@1#eq:generalleft · line 350
$E_{m} P_{m} ψ = ψ - H_{m} [ψ],$
unlabeled · line 340
$ψ ∣_{Σ_{0}} = 0, \partial_{t} ψ ∣_{Σ_{0}} = 0,$

Lemma: The left-inverse identity with its data condition

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