Manuscript IX-Free-2@1 · IX-Free-2.tex · section · abb2397a

An explicit sharp trace witness

Gate S: boundary-charge separation › An explicit sharp trace witness · lines 1075–1101

IX-Free-2@1#eq:advancedtrace

The following trace formula fixes the sign for $□ = - \partial_{t}^{2} + Δ$ :

(G_{A}^{0} a)_{μ} (0, z) = - \frac{1}{4 π} \int_{R^{3}} \frac{a _{μ} ( ∣ x - z ∣ , x )}{∣ x - z ∣} d^{3} x, a \in C_{c}^{\infty} (M_{+}; T^{*} M) .

It is the advanced Kirchhoff formula; compare Ivrii. To see the normalization, the spatial Fourier transform of $δ (s - ∣ x ∣) / (4 π ∣ x ∣)$ , for $s > 0$ , is $sin (s ∣ p ∣) /∣ p ∣$ . The zero-future-data solution of

\partial_{t}^{2} u + ∣ p ∣^{2} u = - a

at $t = 0$ is

- \int_{0}^{\infty} \frac{sin ( s ∣ p ∣ )}{∣ p ∣} a (s, p) d s .

Inverting the spatial transform gives (eq:advancedtrace).

Displayed formulas in this passage

IX-Free-2@1#eq:advancedtrace · line 1079
$(G_{A}^{0} a)_{μ} (0, z) = - \frac{1}{4 π} \int_{R^{3}} \frac{a _{μ} ( ∣ x - z ∣ , x )}{∣ x - z ∣} d^{3} x, a \in C_{c}^{\infty} (M_{+}; T^{*} M) .$
unlabeled · line 1091
$\partial_{t}^{2} u + ∣ p ∣^{2} u = - a$
unlabeled · line 1095
$- \int_{0}^{\infty} \frac{sin ( s ∣ p ∣ )}{∣ p ∣} a (s, p) d s .$

An explicit sharp trace witness

Displayed formulas in this passage

Nearby in this section