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

Normalization cross-check

Continuum defect certification and the next accuracy target › Normalization cross-check · lines 1940–1959

For $c = A + κ B$ , expansion gives

∥ ψ (1) ∥^{2} = 1 + 2 Re [- i \int_{0}^{1} g (- s - 1) c (s) d s] + \int_{0}^{1} \int_{0}^{1} c (s)^{*} g (s - t) c (t) d s d t .

This follows from the exponential packet measure and Fubini. With frequency cutoff $W$ ,

\int_{W}^{\infty} ∣ ψ (1, ω) ∣^{2} d ω \leq e^{- 2 W} (1 + \int_{0}^{1} ∣ c (s) ∣ d s)^{2} .

A finite spectral-system norm near one is not automatically a continuum normalization certificate.

Displayed formulas in this passage

unlabeled · line 1943
$∥ ψ (1) ∥^{2} = 1 + 2 Re [- i \int_{0}^{1} g (- s - 1) c (s) d s] + \int_{0}^{1} \int_{0}^{1} c (s)^{*} g (s - t) c (t) d s d t .$
unlabeled · line 1952
$\int_{W}^{\infty} ∣ ψ (1, ω) ∣^{2} d ω \leq e^{- 2 W} (1 + \int_{0}^{1} ∣ c (s) ∣ d s)^{2} .$