Manuscript Hawk02-C@2 · Hawk02-C.tex · section · bb332d66

Directly checkable packet and chain formulas

Directly checkable packet and chain formulas · lines 2557–2592

The packet coefficients are determined by

b_{n} (T) = - i c e^{- i ω_{0} T} \int_{0}^{T} a_{c} (s) \int_{- 2 c}^{2 c} w_{c} (x) U_{n} (x / (2 c)) e^{- i x (T - s)} d x d s .

They are not independent completion parameters.

On the coupled cyclic space,

e^{- i T (ω_{0} I + K)} ∣ 0 ⟩ = (A (T), b_{0} (T), b_{1} (T), \dots),

and

∣ A (T) ∣^{2} + n = 0 \sum^{\infty} ∣ b_{n} (T) ∣^{2} = 1.

The complete residual retains missed coefficients coherently with the source amplitude.

A finite chain has

(H^{[L]})_{j k} = ω_{0} δ_{j k} + c δ_{j, k + 1} + c δ_{j + 1, k}, 0 \leq j, k \leq L .

Its exact finite endpoint is unitary in that finite model. Its continuum discrepancy is separately bounded by Theorem thm:C7.

Displayed formulas in this passage

unlabeled · line 2560
$b_{n} (T) = - i c e^{- i ω_{0} T} \int_{0}^{T} a_{c} (s) \int_{- 2 c}^{2 c} w_{c} (x) U_{n} (x / (2 c)) e^{- i x (T - s)} d x d s .$
unlabeled · line 2571
$e^{- i T (ω_{0} I + K)} ∣ 0 ⟩ = (A (T), b_{0} (T), b_{1} (T), \dots),$
unlabeled · line 2576
$∣ A (T) ∣^{2} + n = 0 \sum^{\infty} ∣ b_{n} (T) ∣^{2} = 1.$
unlabeled · line 2583
$(H^{[L]})_{j k} = ω_{0} δ_{j k} + c δ_{j, k + 1} + c δ_{j + 1, k}, 0 \leq j, k \leq L .$