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Manuscript IV@1 · IV.tex · figure · 351cb2d7

Finite-band domain argument

Shared conventions and elementary measurement facts › A self-contained quadratic-domain lemma · lines 684–718

IV@1#fig:finite-band-domain

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Model 1 domain argument

00
11
22
⋯\cdots
nn
n+1n+1
n+2n+2

number ladder

quadratic operator changes number by at most two, ∣ΔN∣≤2|\Delta N|\le 2
∥AΨ≤n∥≤C(n+2)∥Ψ≤n∥\lVert A\Psi_{\le n}\rVert\le C(n+2)\lVert\Psi_{\le n}\rVert
∥AjΨ≤n∥≤(2C)j(n+j)!n!∥Ψ≤n∥\lVert A^j\Psi_{\le n}\rVert\le (2C)^j\frac{(n+j)!}{n!}\lVert\Psi_{\le n}\rVert
analytic vectors, common radius 2C∣t∣<12C|t|<1
→
local unitary group
→
vanishing deficiency spaces
→
essential self-adjointness
∥AΨ∥≤C∥(N+2)Ψ∥⟹Dom⁡N⊂Dom⁡A‾\lVert A\Psi\rVert\le C\lVert(\mathsf N+2)\Psi\rVert \Longrightarrow \operatorname{Dom}\mathsf N\subset\operatorname{Dom}\overline A
Not a claim of uncompressed measured dynamics in Models 2 or 3.

Finite-band domain argument. Linear growth of the number-sector bound and number changes of at most two yield analytic finite-particle vectors with a common convergence radius. The resulting local unitary construction eliminates the deficiency spaces. The number graph estimate also gives the stated domain inclusion for the closure.

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