Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

Boundary data and the complete transverse product ledger

Boundary data and the complete transverse product ledger · lines 1382–1413

The radial expression in angular mode $n$ is

- \partial_{r}^{2} - r^{- 1} \partial_{r} + ν_{n}^{2} r^{- 2} in L^{2} ((0, \infty), r d r) .

The deficient modes are

D_{β} = {n : ∣ n + β ∣ < α} .

For $0 < β < 1$ , these can only be $0, - 1$ :

0 \in D_{β} ⟺ β < α, - 1 \in D_{β} ⟺ 1 - β < α .

There are two precisely when $1 - α < β < α$ . If there are no deficient modes, Friedrichs is the unique transverse extension.

For $0 < ν_{j} < 1$ , boundary data are

u_{j} (r) = a_{j} r^{- ν_{j}} + b_{j} r^{ν_{j}} + o (r^{ν_{j}}) .

The integrated Green form is, with one fixed overall sign convention,

2 j \sum ν_{j} (a ˉ_{j} b^{'}_{j} - b ˉ_{j} a^{'}_{j}) .

Its null Lagrangians classify the extensions. Write $N = diag (ν_{j})$ .

Displayed formulas in this passage

unlabeled · line 1386
$- \partial_{r}^{2} - r^{- 1} \partial_{r} + ν_{n}^{2} r^{- 2} in L^{2} ((0, \infty), r d r) .$
unlabeled · line 1391
$D_{β} = {n : ∣ n + β ∣ < α} .$
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$0 \in D_{β} ⟺ β < α, - 1 \in D_{β} ⟺ 1 - β < α .$
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$u_{j} (r) = a_{j} r^{- ν_{j}} + b_{j} r^{ν_{j}} + o (r^{ν_{j}}) .$
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$2 j \sum ν_{j} (a ˉ_{j} b^{'}_{j} - b ˉ_{j} a^{'}_{j}) .$