claimroutine

Polarization is a local angular coordinate

Shows polarization v(ξ)v(\xi) is a local angular coordinate
turns Hessian positivity into a local chart
Claim·§E.1 Eigenvectors and the spatial fiber·inside thmLocal smoothing for symmetric pencil waves

Statement (paper)

Let ω∈Γ\omega\in\Gamma be a unit direction, let e0e_0 be the unit top eigenvector of C(ω)C(\omega), d=n−1d=n-1, and let E(ξ)=(1,v(ξ))tE(\xi)=(1,v(\xi))^{\mathsf t} be the smooth degree-zero real eigenvector near ω\omega. With S⊥(ξ)S_\perp(\xi) the lower d×dd\times d block of S(ξ)=ϕ(ξ)I−C(ξ)S(\xi)=\phi(\xi)I-C(\xi), one has c∣ξ∣I≤S⊥(ξ)≤C∣ξ∣Ic|\xi|I\le S_\perp(\xi)\le C|\xi|I on a small conic neighborhood, and for LwLw the orthogonal projection of C(w)e0C(w)e_0 on e0⊥e_0^\perp, D2ϕ(ω)[w,w]=2(Lw)tS⊥(ω)−1LwD^2\phi(\omega)[w,w]=2(Lw)^{\mathsf t}S_\perp(\omega)^{-1}Lw. Consequently L:ω⊥→e0⊥L:\omega^\perp\to e_0^\perp is an isomorphism, and so is Dv(ω)=S⊥(ω)−1LDv(\omega)=S_\perp(\omega)^{-1}L on ω⊥\omega^\perp, so the polarization vv is a local angular coordinate.

In plain English

Near a direction ω∈Γ\omega\in\Gamma, the top-eigenvalue gap matrix S⊥(ξ)S_\perp(\xi) is uniformly positive definite, and D2ϕ(ω)[w,w]=2(Lw)tS⊥(ω)−1LwD^2\phi(\omega)[w,w]=2(Lw)^{\mathsf t}S_\perp(\omega)^{-1}Lw, where LwLw is the projection of C(w)e0C(w)e_0 onto e0⊥e_0^\perp. Hence L:ω⊥→e0⊥L:\omega^\perp\to e_0^\perp and Dv(ω)=S⊥(ω)−1LDv(\omega)=S_\perp(\omega)^{-1}L are isomorphisms, so the eigenvector polarization vv is a local angular coordinate.

Why it's here

This turns the hypothesis that the transverse Hessian of ϕ\phi is positive definite into the statement that the polarization map ξ↦v(ξ)\xi\mapsto v(\xi) is a local diffeomorphism from directions to the fiber Rd\RR^d. The later fiber-chart claim relies on this to parametrize the eigenvector fiber by WW near v(ω)v(\omega) and to control the geometry through B(v(ξ))=∇ϕ(ξ)B(v(\xi))=\nabla\phi(\xi). The key idea is the second-order perturbation formula for a simple top eigenvalue, whose Hessian is a positive quadratic form in LwLw weighted by S⊥−1S_\perp^{-1}.
Classified routine:
Direct consequence of standard second-order perturbation theory for a simple eigenvalue (Rayleigh–Schrödinger/Kato), using linearity of C(ξ)C(\xi) in ξ\xi, so that D2ϕD^2\phi is the positive quadratic form in LwLw weighted by S⊥−1S_\perp^{-1}, plus the inverse function theorem.

Uses

Nothing else in the paper

Used by

claimFiber chart and geometric bounds

Sources

PaperOpen
Section
§E.1 Eigenvectors and the spatial fiber
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sections/pencil.tex:52
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