Manuscript XI-B@1 · XI-B.tex · section · 063b989e

Wave inverses on the slab

Causal Green operators and their normalization › Wave inverses on the slab · lines 582–600

XI-B@1#eq:ER, XI-B@1#eq:EA

For $μ \geq 0$ , let $Ω_{μ} = - Δ + μ^{2}$ . For a smooth source $F$ define

(E_{μ}^{R} F) (t) (E_{μ}^{A} F) (t) = \int_{0}^{t} \frac{sin (( t - s ) Ω _{μ} )}{Ω _{μ}} F (s) d s, = \int_{t}^{T} \frac{sin (( s - t ) Ω _{μ} )}{Ω _{μ}} F (s) d s .

The quotient has its continuous zero-frequency value. For nondecaying smooth sources these formulas are interpreted locally by finite propagation. On compact interior sources they are the restrictions of the full Minkowski retarded and advanced operators.

Displayed formulas in this passage

XI-B@1#eq:ER · line 586
$(E_{μ}^{R} F) (t) (E_{μ}^{A} F) (t) = \int_{0}^{t} \frac{sin (( t - s ) Ω _{μ} )}{Ω _{μ}} F (s) d s, = \int_{t}^{T} \frac{sin (( s - t ) Ω _{μ} )}{Ω _{μ}} F (s) d s .$
XI-B@1#eq:EA · line 586
$(E_{μ}^{R} F) (t) (E_{μ}^{A} F) (t) = \int_{0}^{t} \frac{sin (( t - s ) Ω _{μ} )}{Ω _{μ}} F (s) d s, = \int_{t}^{T} \frac{sin (( s - t ) Ω _{μ} )}{Ω _{μ}} F (s) d s .$

Wave inverses on the slab

Displayed formulas in this passage

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