Manuscript En02-E@2 · En02-E-v2.tex · lemma · c2769986

Lemma: Conservative amplitude bound

Radial boundary-error estimate · lines 1425–1460 · proof included or linked

En02-E@2#lem:josterror

For $ω \geq ω_{-} > 0$ ,

x sup ∣ f_{V} - f_{W} ∣ \leq \frac{d}{ω _{-}} e^{(v + w) / ω_{-}} .

For incoming and reflected coefficients $a, b$ ,

∣ a_{V} - a_{W} ∣, ∣ b_{V} - b_{W} ∣ \leq A = \frac{d}{2 ω _{-}} e^{(v + w) / ω_{-}} (1 + \frac{w}{ω _{-}}) .

Consequently

∣ τ_{V} - τ_{W} ∣ \leq A, ∣ r_{V} - r_{W} ∣ \leq 2 A, ∥ S_{V} - S_{W} ∥_{\infty} \leq min {2, 4 A} .

Proof. Volterra iteration gives $sup ∣ f_{W} ∣ \leq e^{w / ω_{-}}$ . Subtracting equations and applying the Volterra resolvent bound $e^{v / ω_{-}}$ gives the solution difference.

The coefficient integrals contain $(2 i ω)^{- 1} \int e^{\pm i ω y} V f_{V} d y$ . Write $V f_{V} - W f_{W} = (V - W) f_{V} + W (f_{V} - f_{W})$ to obtain $A$ . Current conservation gives $∣ a ∣ \geq 1$ , $τ = 1/ a$ , and $∣ b / a ∣ \leq 1$ ; hence the stated transmission and reflection bounds. Apply the analogous left Jost construction for opposite incidence. The four-entry Frobenius error is at most $10 A \leq 4 A$ ; unitarity separately bounds it by two.

Displayed formulas in this passage

unlabeled · line 1428
$x sup ∣ f_{V} - f_{W} ∣ \leq \frac{d}{ω _{-}} e^{(v + w) / ω_{-}} .$
unlabeled · line 1433
$∣ a_{V} - a_{W} ∣, ∣ b_{V} - b_{W} ∣ \leq A = \frac{d}{2 ω _{-}} e^{(v + w) / ω_{-}} (1 + \frac{w}{ω _{-}}) .$
unlabeled · line 1440
$∣ τ_{V} - τ_{W} ∣ \leq A, ∣ r_{V} - r_{W} ∣ \leq 2 A, ∥ S_{V} - S_{W} ∥_{\infty} \leq min {2, 4 A} .$

Lemma: Conservative amplitude bound

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