Manuscript XVII-A@1 · XVII-A.tex · proposition · 711d7ee4

Proposition: Actual boundary limit

Boundary data and the declared representation correspondence › The correspondence is declared, not inferred · lines 386–410 · proof included or linked

XVII-A@1#prop:boundary-limit, XVII-A@1#eq:boundary-limit

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For every $t$ ,

r ↓ 0 lim r^{- 1} Φ (t, π /2 - r) = U^{†} β (t) U,

in operator norm. The convergence is uniform on compact $t$ sets.

Proof. For each $n$ ,

sin (n (π - r)) = (- 1)^{n + 1} sin (n r), r^{- 1} sin (n r) ⟶ n .

Multiplication by $1/ π n$ gives precisely $c_{n}$ . The sum is finite, the oscillator matrices are bounded, and the temporal phases have modulus one. These facts give norm convergence and the stated uniformity.

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XVII-A@1#eq:boundary-limit · line 389
$r ↓ 0 lim r^{- 1} Φ (t, π /2 - r) = U^{†} β (t) U,$
unlabeled · line 400
$sin (n (π - r)) = (- 1)^{n + 1} sin (n r), r^{- 1} sin (n r) ⟶ n .$

Proposition: Actual boundary limit

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