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

The coupled autonomous chain

Finite-interface and predictive bridges › The coupled autonomous chain · lines 1474–1509

Hawk02-C@2#eq:chain

The coupled cyclic space is

H_{cyc} = C \oplus L^{2} (I_{c}, d ν) .

Its orthogonal field complement evolves freely and remains unoccupied on the ideal preparation.

Since

(ν - ω_{0}) ϕ_{n} = c ϕ_{n + 1} + c ϕ_{n - 1}, g = c ϕ_{0},

the ordered source-plus-packet basis gives

h_{g}_{H_{cyc}} = ω_{0} I + K, K = c (S + S^{†}) on ℓ^{2} (N_{0}),

where $S ∣ n ⟩ = ∣ n + 1 ⟩$ and $∥ K ∥ \leq 2 c$ .

Let $K^{[N]}$ be the open chain on nodes $0, \dots, N$ , with nearest-neighbor hopping $c$ . Its Hamiltonian $ω_{0} I + K^{[N]}$ is fixed by the form factor and packet recurrence, not by the calculated endpoint.

Define the uncapped series

S_{N} (T) = 2 m = N + 1 \sum^{\infty} \frac{( 2 c T ) ^{m}}{m !}, ζ_{N} (T) = min {1, S_{N} (T)} .

Displayed formulas in this passage

Hawk02-C@2#eq:chain · line 1490
$h_{g}_{H_{cyc}} = ω_{0} I + K, K = c (S + S^{†}) on ℓ^{2} (N_{0}),$
unlabeled · line 1477
$H_{cyc} = C \oplus L^{2} (I_{c}, d ν) .$
unlabeled · line 1485
$(ν - ω_{0}) ϕ_{n} = c ϕ_{n + 1} + c ϕ_{n - 1}, g = c ϕ_{0},$
unlabeled · line 1504
$S_{N} (T) = 2 m = N + 1 \sum^{\infty} \frac{( 2 c T ) ^{m}}{m !}, ζ_{N} (T) = min {1, S_{N} (T)} .$

The coupled autonomous chain

Displayed formulas in this passage

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