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

Theorem: Resolvent correction on every boundary chart

Krein formula, infrared control, and nonnegative states · lines 1475–1542 · proof included or linked

I-B-VP@3#thm:krein, I-B-VP@3#eq:CW, I-B-VP@3#eq:resolvent, I-B-VP@3#eq:Cfinite

For the extension (eq:relation), define

C_{W} (p) = \frac{1}{2} E^{- 1} V [H_{W} + V^{†} Δ (p) V]^{- 1} V^{†} E^{- 1} .

At values where the inverse exists, the transverse resolvent difference is

(B_{⊥, W} + p^{2})^{- 1} (x, x^{'}) - (B_{⊥, F} + p^{2})^{- 1} (x, x^{'}) = j k \sum ψ_{j} (φ) K_{ν_{j}} (p r) C_{W, j k} (p) K_{ν_{k}} (p r^{'}) ψ_{k} (φ^{'}) .

In the full finite chart,

C (p) = \frac{1}{2} E^{- 1} (H + Δ)^{- 1} E^{- 1} = (2 E H E + S)^{- 1} .

For a diagonal mode,

c_{λ} (p) = \frac{c _{0} ( ν )}{1 + λ \frac{Γ ( 1 + ν )}{Γ ( 1 - ν )} ( p /2 ) ^{- 2 ν}} .

For $E_{0}$ , $c = c_{0} (ν)$ . For $L_{L}$ ,

c_{L} (p) = \frac{1}{lo g ( p / κ _{L} )}, κ_{L} = \frac{2 e ^{- γ_{E}}}{L} .

Proof. The resolvent difference is a finite sum of decaying homogeneous solutions $K_{ν_{j}} (p r)$ in each slot. Near zero,

K_{ν} (p r) = e_{ν} (p) r^{- ν} - s_{ν} d_{ν} (p) r^{ν} + o (r^{ν}) .

For a source column $w$ , the Friedrichs term contributes $D w$ to the regular boundary coefficient. Thus

a = E C w, b = D w - D S C w .

Set $a = V ζ$ . Since $N D = \frac{1}{2} E^{- 1}$ and $N D S E^{- 1} = Δ$ , condition (eq:relation) becomes

[H_{W} + V^{†} Δ V] ζ = \frac{1}{2} V^{†} E^{- 1} w .

Then $C w = E^{- 1} V ζ$ , proving (eq:CW). For the full chart use $2 E Δ E = S$ . The diagonal case follows immediately.

For $ν = 0$ , the expansion $K_{0} (p r) = - lo g (p r /2) - γ_{E} + o (1)$ gives $c_{L}^{- 1} = lo g (p L /2) + γ_{E}$ .

Displayed formulas in this passage

I-B-VP@3#eq:CW · line 1478
$C_{W} (p) = \frac{1}{2} E^{- 1} V [H_{W} + V^{†} Δ (p) V]^{- 1} V^{†} E^{- 1} .$
I-B-VP@3#eq:resolvent · line 1485
$(B_{⊥, W} + p^{2})^{- 1} (x, x^{'}) - (B_{⊥, F} + p^{2})^{- 1} (x, x^{'}) = j k \sum ψ_{j} (φ) K_{ν_{j}} (p r) C_{W, j k} (p) K_{ν_{k}} (p r^{'}) ψ_{k} (φ^{'}) .$
I-B-VP@3#eq:Cfinite · line 1495
$C (p) = \frac{1}{2} E^{- 1} (H + Δ)^{- 1} E^{- 1} = (2 E H E + S)^{- 1} .$
unlabeled · line 1501
$c_{λ} (p) = \frac{c _{0} ( ν )}{1 + λ \frac{Γ ( 1 + ν )}{Γ ( 1 - ν )} ( p /2 ) ^{- 2 ν}} .$
unlabeled · line 1509
$c_{L} (p) = \frac{1}{lo g ( p / κ _{L} )}, κ_{L} = \frac{2 e ^{- γ_{E}}}{L} .$
unlabeled · line 1519
$K_{ν} (p r) = e_{ν} (p) r^{- ν} - s_{ν} d_{ν} (p) r^{ν} + o (r^{ν}) .$
unlabeled · line 1525
$a = E C w, b = D w - D S C w .$
unlabeled · line 1530
$[H_{W} + V^{†} Δ V] ζ = \frac{1}{2} V^{†} E^{- 1} w .$