TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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CTA complex geometrical optics solvability and remainder bound

Statement

Assume the Axiom of Choice and Countable Choice.

Let $(M, g)$ be a transversally anisotropic product domain, let $q \in C^{\infty} (\overline{M}, R)$ , and let $ψ \in C^{\infty}$ be real on a neighborhood of the transversal projection of $M$ , with $∣ d ψ ∣_{g_{0}} = 1$ . Suppose $a \in C^{\infty} (\overline{M}, C)$ satisfies $2 \partial_{x_{1}} a + 2 i g_{0} (\nabla_{g_{0}} ψ, \nabla_{g_{0}} a) + i (Δ_{g_{0}} ψ) a = 0.$ Then, for all sufficiently large $τ$ , there is an exact solution $u_{τ} = e^{- τ (x_{1} + i ψ)} a + e^{- τ x_{1}} r_{τ}, (- Δ_{g} + q) u_{τ} = 0 in M^{\circ},$ with $r_{τ} \in H^{1} (M)$ and $∥ r_{τ} ∥_{L^{2} (M)} \leq C τ^{- 1} ∥ (- Δ_{g} + q) a ∥_{L^{2} (M)}$ . No boundary value is prescribed for $r_{τ}$ .

Statement references

Product metric and dimension

Product Carleman estimate and L2 right solvability

Facts & Assumptions

[A1]

The Axiom of Choice is used for the Hahn--Banach extension in the product Carleman right solver [F2].

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is used for its Hilbert-space Riesz representation [F2].

Given: the product domain, smooth real $q$ , real eikonal function $ψ$ with unit transversal gradient, and smooth transport amplitude $a$ .

[F1]

Transversally anisotropic product domain: $g = d x_{1}^{2} \oplus g_{0}$ and $Δ_{g} = \partial_{x_{1}}^{2} + Δ_{g_{0}}$ .

[F2]

Limiting Carleman estimate and right solvability on a product domain: for $φ = x_{1}$ , the conjugated operator $P_{φ, h, q} = e^{φ / h} h^{2} (- Δ_{g} + q) e^{- φ / h}$ has a distributional right inverse on $L^{2}$ with norm at most $C h^{- 1}$ .

[F3]

Riemannian gradient and Laplace–Beltrami operator as the trace of the Hessian: the product metric extends the gradient bilinearly to complex phases, and $Δ_{g} = {div}_{g} \nabla_{g}$ .

[F4]

External input: Dos Santos Ferreira--Kurylev--Lassas--Salo, Proposition 2.1 (PDF p. 10), for a compact oriented CTA manifold, bounded potential, either sign of sufficiently large real $τ$ , and arbitrary $f \in L^{2}$ , gives $r \in H^{1}$ solving $e^{τ x_{1}} (- Δ_{g} + q) e^{- τ x_{1}} r = f$ , with $∥ r ∥_{H^{α}} \leq C ∣ τ ∣^{α - 1} ∥ f ∥_{2}$ for $0 \leq α \leq 1$ , with no prescribed boundary value. We import this statement, not a reconstructed shifted Carleman proof. Its statement and the application in Propositions 2.1--2.2, PDF pp. 10--11, were inspected; the older Proposition 4.4 citation issue is not treated as a local proof.

[F5]

The orientation restriction in [F4] is removed for scalar equations by the orientation double cover of $M_{0}$ . Lift $M, q, f$ to its product cover, solve there, and average the solution with its deck transform. The invariant solution descends to $H^{1} (M)$ and solves the original equation. The degree-two factors in the squared $L^{2}$ norms cancel in the estimate; the same applies to the local Sobolev norms. A disconnected cover is handled componentwise, using only finitely many choices.

Proof

Given: the data in the statement.

1.1F1F3givenalgebra

Put $Φ = x_{1} + i ψ$ . By [F1] and $∣ d ψ ∣_{g_{0}} = 1$ , $g (d Φ, d Φ) = 1 + i^{2} ∣ d ψ ∣_{g_{0}}^{2} = 0.$ Direct conjugation gives $(- Δ_{g} + q) (e^{- τ Φ} a) = e^{- τ Φ} [- τ^{2} g (d Φ, d Φ) a + τ (2 g (\nabla_{g} Φ, \nabla_{g} a) + (Δ_{g} Φ) a) + (- Δ_{g} + q) a] .$ The $τ^{2}$ term vanishes by the eikonal equality, and the coefficient of $τ$ is exactly the transport equation in the statement. Hence the residual is $e^{- τ Φ} (- Δ_{g} + q) a$ .

2.1A1A2F4F5step 1.1algebra

Set $h = τ^{- 1}$ and conjugate by the real weight $e^{- τ x_{1}}$ . The equation for the correction is $P_{x_{1}, h, q} r_{τ} = - h^{2} e^{- i τ ψ} (- Δ_{g} + q) a .$ The right side is in $L^{2}$ and has norm $h^{2} ∥ (- Δ_{g} + q) a ∥_{2}$ . By [F4]--[F5], with the equation divided by $h^{2}$ , there is an $H^{1}$ correction (and hence a distributional) $r_{τ} \in L^{2} (M)$ with $∥ r_{τ} ∥_{2} \leq C h^{- 1} h^{2} ∥ (- Δ_{g} + q) a ∥_{2} = C τ^{- 1} ∥ (- Δ_{g} + q) a ∥_{2} .$ Thus the asserted remainder estimate and the equation hold distributionally.

3.1F4F5step 1.1step 2.1∎

Choose the $H^{1}$ correction from step 2.1. Multiplication by the smooth exponentials preserves $H^{1} (M)$ for each fixed $τ$ , and the correction equation cancels the residual computed in step 1.1. Thus $u_{τ}$ is exact, has the stated remainder bound, and has no imposed correction boundary value. The additional boundary regularity is the external input [F4], rather than a consequence asserted for the $L^{2}$ solver [F2].

Depends on

Used by

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Sources

Dos Santos Ferreira, Kenig, Salo, and Uhlmann, Limiting Carleman weights and anisotropic inverse problems (2009) (standard reference, not scraped)
Dos Santos Ferreira, Kurylev, Lassas, and Salo, The Calderon problem in transversally anisotropic geometries (2016) (standard reference, not scraped)
V. A. Sharafutdinov, Integral Geometry of Tensor Fields (1994) (standard reference, not scraped)