Manuscript Em03-H@8 · Em03-H-v8.tex · theorem · 91e9e3bd

Theorem: Stored-packet Volterra characterization

Restricted continuum experiment › Exact one-species evolution · lines 1221–1249 · proof included or linked

Em03-H@8#thm:volterra

The exact evolution is equivalent to the unique continuous solution

A (u) B (u) = g (u + t_{0}) - i \int_{0}^{u} g (u - s) [v (s) A (s) + κ (s) B (s)] d s, = - i \int_{0}^{u} e^{- i (u - s)} κ (s) A (s) d s .

The matter packet is

ψ (u, ω) = f (ω) e^{- i ω (u + t_{0})} - i f (ω) \int_{0}^{u} e^{- i ω (u - s)} [v (s) A (s) + κ (s) B (s)] d s .

For the primary experiment, $A (0) = (4 - 2 i) /5$ , $B (0) = 0$ .

Proof. Variation of constants gives the packet formula. Contracting with $f$ uses $⟨ f, e^{- i t h} f ⟩ = g (t)$ . Variation of constants for $i B ˙ = B + κ A$ gives the second equation. Conversely, the packet formula reconstructs the mild Schrödinger solution; uniqueness proves equivalence.

Displayed formulas in this passage

unlabeled · line 1224
$A (u) B (u) = g (u + t_{0}) - i \int_{0}^{u} g (u - s) [v (s) A (s) + κ (s) B (s)] d s, = - i \int_{0}^{u} e^{- i (u - s)} κ (s) A (s) d s .$
unlabeled · line 1233
$ψ (u, ω) = f (ω) e^{- i ω (u + t_{0})} - i f (ω) \int_{0}^{u} e^{- i ω (u - s)} [v (s) A (s) + κ (s) B (s)] d s .$

Theorem: Stored-packet Volterra characterization

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