Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

Krein formula, infrared control, and nonnegative states

Krein formula, infrared control, and nonnegative states · lines 1457–1474

For $0 < ν < 1$ , set

s_{ν} = \frac{π}{2 sin π ν}, e_{ν} (p) = s_{ν} \frac{( p /2 ) ^{- ν}}{Γ ( 1 - ν )}, d_{ν} (p) = \frac{( p /2 ) ^{ν}}{Γ ( 1 + ν )},

Δ_{ν} (p) = ν \frac{Γ ( 1 - ν )}{Γ ( 1 + ν )} (p /2)^{2 ν}, c_{0} (ν) = \frac{2 sin π ν}{π} .

Let $E, D, Δ, S$ be the corresponding diagonal matrices, with $S_{j j} = s_{ν_{j}}$ . In particular $d_{ν} e_{ν} = 1/ (2 ν)$ .

Displayed formulas in this passage

unlabeled · line 1461
$s_{ν} = \frac{π}{2 sin π ν}, e_{ν} (p) = s_{ν} \frac{( p /2 ) ^{- ν}}{Γ ( 1 - ν )}, d_{ν} (p) = \frac{( p /2 ) ^{ν}}{Γ ( 1 + ν )},$
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$Δ_{ν} (p) = ν \frac{Γ ( 1 - ν )}{Γ ( 1 + ν )} (p /2)^{2 ν}, c_{0} (ν) = \frac{2 sin π ν}{π} .$