Manuscript En07-C@1 · En07-C.tex · proposition · 3f7e9ebb

Proposition: A fixed autonomous unitary does not supply a nontrivial norm endpoint

Access-preserving endpoint certificates › Endpoint topology and finite autonomous evolution · lines 1726–1761 · proof included or linked

En07-C@1#prop:autonomous

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Let $H$ be a finite-dimensional time-independent Hamiltonian, $P_{0} : C \to H_{tot}$ an isometry, and

V_{t} = e^{- i t H /ℏ} P_{0}

in a fixed output identification. If $V_{t}$ has an operator-norm limit $V_{*}$ as $t \to \infty$ , then $V_{t} = P_{0} = V_{*}$ for every $t$ , and $H P_{0} = 0$ . This proposition does not cover a finite terminal time, a time-dependent dynamics, or a separately justified time-dependent output comparison.

Proof. For every $s \geq 0$ ,

e^{- i sH /ℏ} V_{*} = t \to \infty lim V_{t + s} = V_{*} .

Therefore

∥ V_{t} - V_{*} ∥_{\infty} = e^{- i t H /ℏ} (P_{0} - V_{*})_{\infty} = ∥ P_{0} - V_{*} ∥_{\infty} .

The left side tends to zero, so $P_{0} = V_{*}$ . The invariance under every $s$ then gives $H P_{0} = 0$ .

Displayed formulas in this passage

unlabeled · line 1731
$V_{t} = e^{- i t H /ℏ} P_{0}$
unlabeled · line 1744
$e^{- i sH /ℏ} V_{*} = t \to \infty lim V_{t + s} = V_{*} .$
unlabeled · line 1752
$∥ V_{t} - V_{*} ∥_{\infty} = e^{- i t H /ℏ} (P_{0} - V_{*})_{\infty} = ∥ P_{0} - V_{*} ∥_{\infty} .$

Proposition: A fixed autonomous unitary does not supply a nontrivial norm endpoint

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