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

Proposition: Analytical coefficient and tail formulas

Finite-interface and predictive bridges › Exact continuum chain coefficients · lines 1644–1690 · proof included or linked

Hawk02-C@2#prop:coefficients, Hawk02-C@2#eq:chaincoeff, Hawk02-C@2#eq:exactell

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For $T > 0$ , the coupled chain coefficients are

v_{n} (T) = e^{- i ω_{0} T} (- i)^{n} \frac{n + 1}{c T} J_{n + 1} (2 c T), n \geq 0,

with $v_{0} = A$ and $b_{n} = v_{n + 1}$ . Consequently,

ℓ_{N} (T) = n = N + 1 \sum^{\infty} \frac{( n + 1 ) ^{2}}{c ^{2} T ^{2}} J_{n + 1} (2 c T)^{2} .

These are continuum formulas, not values obtained from the reported finite-chain simulation.

Proof. The spectral transformation of the half-line chain sends $∣ n ⟩$ to $U_{n} (x / (2 c))$ in $L^{2} (w_{c} (x) d x)$ . Its unitarity follows from polynomial orthogonality and density; the recurrence intertwines $K$ with multiplication by $x$ . Hence

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

With $x = 2 c cos θ$ , the integral is

\frac{1}{π} \int_{0}^{π} [cos (n θ) - cos ((n + 2) θ)] e^{- i (2 c T) c o s θ} d θ .

The integer-order Bessel integral and recurrence DLMF (Sections 10.6 and 10.9) give

(- i)^{n} [J_{n} (2 c T) + J_{n + 2} (2 c T)] = (- i)^{n} \frac{n + 1}{c T} J_{n + 1} (2 c T) .

Parseval's identity and the packet indexing give (eq:exactell).

Displayed formulas in this passage

Hawk02-C@2#eq:chaincoeff · line 1647
$v_{n} (T) = e^{- i ω_{0} T} (- i)^{n} \frac{n + 1}{c T} J_{n + 1} (2 c T), n \geq 0,$
Hawk02-C@2#eq:exactell · line 1655
$ℓ_{N} (T) = n = N + 1 \sum^{\infty} \frac{( n + 1 ) ^{2}}{c ^{2} T ^{2}} J_{n + 1} (2 c T)^{2} .$
unlabeled · line 1671
$v_{n} (T) = e^{- i ω_{0} T} \int_{- 2 c}^{2 c} w_{c} (x) U_{n} (x / (2 c)) e^{- i x T} d x .$
unlabeled · line 1676
$\frac{1}{π} \int_{0}^{π} [cos (n θ) - cos ((n + 2) θ)] e^{- i (2 c T) c o s θ} d θ .$
unlabeled · line 1684
$(- i)^{n} [J_{n} (2 c T) + J_{n + 2} (2 c T)] = (- i)^{n} \frac{n + 1}{c T} J_{n + 1} (2 c T) .$