Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · section · f4bd2b11

Definition and compact-test finiteness

The regular host and the spectral covariance › Definition and compact-test finiteness · lines 326–350

I-B-VP-2@1#eq:W

Put $H_{β} = B_{β}$ . The covariance is

W_{β} (T; x, x^{'}) = [\frac{e ^{- i T H_{β}}}{2 H _{β}}] (x, x^{'}), T = t - t^{'} .

For compact spacetime sections,

K_{β} f = \int_{R} e^{i t H_{β}} (2 H_{β})^{- 1/2} f (t) d t, W_{β} (f, g) = ⟨ K_{β} f, K_{β} g ⟩ .

For a real neutral scalar take $β = 0$ . For a charged scalar use the particle and antiparticle sectors on the bundle and its conjugate, as in I-B-VP-v2 :state CheungIBVP.

For completeness, the estimate needed at zero energy is recalled.

Displayed formulas in this passage

I-B-VP-2@1#eq:W · line 330
$W_{β} (T; x, x^{'}) = [\frac{e ^{- i T H_{β}}}{2 H _{β}}] (x, x^{'}), T = t - t^{'} .$
unlabeled · line 337
$K_{β} f = \int_{R} e^{i t H_{β}} (2 H_{β})^{- 1/2} f (t) d t, W_{β} (f, g) = ⟨ K_{β} f, K_{β} g ⟩ .$

Definition and compact-test finiteness

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