Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

Edge removal and its topology

Model 1: the fixed TT plane and removable edge › Edge removal and its topology · lines 660–682

IV-B@1#eq:edge-vector

Choose smooth $0 \leq ϑ \leq 1$ , with $ϑ (s) = 0$ for $s \leq 0$ and $ϑ (s) = 1$ for $s \geq 1$ . Replace

1_{{s \geq t_{0}}} b_{a, f} (s) by ϑ ((s - t_{0}) / ε) b_{a, f} (s),

and denote the resulting operator by $A_{ε}^{(1)}$ . Formula (eq:oneparticle) and integration over an interval of length $ε$ give

∥ u_{a, ε} - u_{a} ∥ \leq C_{a} ε .

Consequently,

(A_{ε}^{(1)} - A^{(1)}) Ψ \leq D ε ∥ (N + 2) Ψ ∥,

with uniform sector bounds for $A_{ε}^{(1)}$ .

Displayed formulas in this passage

IV-B@1#eq:edge-vector · line 672
$∥ u_{a, ε} - u_{a} ∥ \leq C_{a} ε .$
unlabeled · line 664
$1_{{s \geq t_{0}}} b_{a, f} (s) by ϑ ((s - t_{0}) / ε) b_{a, f} (s),$
unlabeled · line 677
$(A_{ε}^{(1)} - A^{(1)}) Ψ \leq D ε ∥ (N + 2) Ψ ∥,$