Manuscript V@1 · V.tex · figure · e7ddf37d

Switched identity and unswitched remainder

The one-oscillator mixed insertion › Number graph and essential self-adjointness · lines 2886–2917

V@1#fig:switched-remainder

metricscalarrepresented quantumdefect / unresolvedsupport / scope

Switched

compact

g

, with

t_0

before its support

R_{1,1}(V(g),\phi(f))=\widehat B_f^{g,t_0}

L_{ga}

includes derivatives of

g

Unswitched

\widehat B_f^{0,t_0}=Q\Phi(v_e^0)+\Pi\Phi(v_r^0)

\mathcal R_{g,f}-\widehat B_f^{0,t_0}=Q\Phi(v_e-v_e^0)+\Pi\Phi(v_r-v_r^0)

Fixed-number bound. Switching and initial-time remainder, not a renormalization ambiguity.

replace

L_{ga}

gL_a

missing term

-\sum_a O_a\int u_f(\partial_i g)\,a^{ij}\partial_j\phi

Switched identity and unswitched comparison. The first retarded coefficient equals the quantized switched Duhamel bilinear when the initial time precedes the vertex support. Comparison with the unswitched response has an explicit mode-vector remainder. Derivatives of the switch are essential, and the remainder is not an extension ambiguity.

Switched identity and unswitched remainder

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