Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · section · 8a9d0682

Euclidean kernel and analytic remainder

Prescribed geometry, bundle, and spectral state › Euclidean kernel and analytic remainder · lines 448–471

I-B-VP-0@1#eq:localHadamard

The spectral continuation

\frac{e ^{- a B_{β}}}{2 B _{β}}, a > 0, a ⟶ ϵ + i (t - t^{'})

gives (eq:W). Continuing the analytic remainder in Proposition prop:kernel gives, on a sufficiently small Lorentzian neighbourhood,

W_{β} (X, X^{'}) = \frac{1}{4 π ^{2} ( ∣ X - X ^{'} ∣ ^{2} - ( t - t ^{'} - i 0 ) ^{2} )} + H_{β} (X, X^{'}),

with smooth $H_{β}$ . This is the local Hadamard property proved in I-B-VP-v2 :hadamard CheungIBVP. We use only this local statement. A global microlocal-spectrum characterization and behaviour at $W$ are not established here.

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I-B-VP-0@1#eq:localHadamard · line 458
$W_{β} (X, X^{'}) = \frac{1}{4 π ^{2} ( ∣ X - X ^{'} ∣ ^{2} - ( t - t ^{'} - i 0 ) ^{2} )} + H_{β} (X, X^{'}),$
unlabeled · line 451
$\frac{e ^{- a B_{β}}}{2 B _{β}}, a > 0, a ⟶ ϵ + i (t - t^{'})$

Euclidean kernel and analytic remainder

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