Manuscript Em02-E@1 · Em02-E-v1.tex · lemma · f3795050

Lemma: Laplacian coefficients

Clock slices and harmonic spatial labels › An explicit Laplace operator expansion · lines 525–574 · proof included or linked

Em02-E@1#lem:laplacian, Em02-E@1#eq:P1, Em02-E@1#eq:P2, Em02-E@1#eq:B

Back to manuscript

One has

Δ_{q_{λ}} = a^{- 2} (Δ + λ P_{1} + \frac{λ ^{2}}{2} P_{2}),

where

P_{1} F P_{2} F = \partial_{i} (A_{1}^{ij} \partial_{j} F) - b_{1} Δ F, = \partial_{i} (A_{2}^{ij} \partial_{j} F) - 2 b_{1} \partial_{i} (A_{1}^{ij} \partial_{j} F) + (2 b_{1}^{2} - b_{2}) Δ F .

In particular,

P_{1} x^{i} = \frac{1}{2} \partial_{i} tr Q_{1} - \partial_{j} Q_{1}^{j i} .

Proof. The determinant and inverse expansions are

\frac{det q _{λ}}{a ^{3}} = 1 + λ b_{1} + \frac{λ ^{2}}{2} b_{2},

a^{2} q_{λ}^{- 1} = 1 - λ Q_{1} + \frac{λ ^{2}}{2} (2 Q_{1}^{2} - Q_{2}) .

Their product has coefficients $A_{1}, A_{2}$ . The inverse determinant factor is $1 - λ b_{1} + λ^{2} (2 b_{1}^{2} - b_{2}) /2$ . Multiplication proves (eq:P1)–(eq:P2). Applying $P_{1}$ to $x^{i}$ gives (eq:B).

Displayed formulas in this passage

Em02-E@1#eq:P1 · line 535
$P_{1} F P_{2} F = \partial_{i} (A_{1}^{ij} \partial_{j} F) - b_{1} Δ F, = \partial_{i} (A_{2}^{ij} \partial_{j} F) - 2 b_{1} \partial_{i} (A_{1}^{ij} \partial_{j} F) + (2 b_{1}^{2} - b_{2}) Δ F .$
Em02-E@1#eq:P2 · line 535
$P_{1} F P_{2} F = \partial_{i} (A_{1}^{ij} \partial_{j} F) - b_{1} Δ F, = \partial_{i} (A_{2}^{ij} \partial_{j} F) - 2 b_{1} \partial_{i} (A_{1}^{ij} \partial_{j} F) + (2 b_{1}^{2} - b_{2}) Δ F .$
Em02-E@1#eq:B · line 548
$P_{1} x^{i} = \frac{1}{2} \partial_{i} tr Q_{1} - \partial_{j} Q_{1}^{j i} .$
unlabeled · line 528
$Δ_{q_{λ}} = a^{- 2} (Δ + λ P_{1} + \frac{λ ^{2}}{2} P_{2}),$
unlabeled · line 558
$\frac{det q _{λ}}{a ^{3}} = 1 + λ b_{1} + \frac{λ ^{2}}{2} b_{2},$
unlabeled · line 563
$a^{2} q_{λ}^{- 1} = 1 - λ Q_{1} + \frac{λ ^{2}}{2} (2 Q_{1}^{2} - Q_{2}) .$

Nearby in this section

section: An explicit Laplace operator expansion