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Well-definedness of the small-norm fixed-frequency scattering map

Statement

Assume Countable Choice and the hypotheses and notation of the definition of fixed-frequency far-field data. For each $q \in U_{D, k}$ there is an $ε > 0$ such that $∥ q^{'} - q ∥_{\infty} < ε$ implies $q^{'} \in U_{D, k}$ ; in particular this metric neighborhood condition holds at zero. For every $q \in U_{D, k}$ and every $d, θ \in S^{2}$ , the solution $u_{q, k, d}$ exists uniquely in $L^{2} (D; C)$ , the integrals defining $S_{k} (q) (θ, d)$ and $B_{k} q (θ, d)$ are absolutely convergent and independent of representatives, and $S_{k} (q) (\cdot, d)$ is continuous. If $R_{0} > \sup_{y \in D} ∣ y ∣$ and $r \geq 2 R_{0}$ , then $\sup_{θ \in S^{2}} ∣ r e^{- i k r} V_{k} u_{q, k, d} (r θ) - S_{k} (q) (θ, d) ∣ \leq \frac{2 R_{0} + ∣ k ∣ R_{0}^{2}}{4 π r} ∥ q ∥_{\infty} \sqrt{M} ∥ u_{q, k, d} ∥_{2} .$ Thus $S_{k} (q) (\cdot, d)$ is the uniform outgoing far-field amplitude in the stated convention.

Statement references

The small-norm domain, volume operator, incident plane wave, scattering data, and first Born data family.

The volume model uses equivalence classes in complex L2 and the stated outgoing Green kernel.

The far-field map is continuous and has the uniform outgoing-wave error bound.

The volume operator is linear and satisfies the Hilbert–Schmidt norm bound used to show openness.

For operator norm strictly below one, the Lippmann–Schwinger equation has a unique L2 solution.

Complex L2 Cauchy–Schwarz and representative-independent pairings.

An L-infinity representative is bounded almost everywhere by its essential norm.

The L1 Fourier transform is defined at every frequency and is representative-independent.

Countable Choice is the declared axiom branch of the volume and L2 suppliers.

Facts & Assumptions

Given: Countable Choice; a bounded Lebesgue-measurable $D \subset R^{3}$ with $0 < M : = ∣ D ∣ < \infty$ ; $k > 0$ ; a contrast $q \in L^{\infty} (D; C)$ ; and directions $d, θ \in S^{2}$ .

[F1]

The volume model defines $K_{q, k}$ on complex $L^{2} (D)$ by the outgoing kernel, treats inputs as equivalence classes, sets the diagonal representative to zero, and uses $∣ q ∣ \leq ∥ q ∥_{\infty}$ almost everywhere (The three-dimensional volume integral scattering model).

[F2]

For $diam (D) < \infty$ , $∥ K_{q, k} ∥_{L^{2} \to L^{2}} \leq C_{D} ∥ q ∥_{\infty}$ , where $C_{D} = \sqrt{M diam (D) / (4 π)}$ (The volume integral operator is well defined and bounded on L2).

[F3]

The plane wave has modulus one, so its restriction to $D$ is in $L^{2} (D)$ with norm $\sqrt{M}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F4]

If $ρ_{k} (q) < 1$ and $f \in L^{2} (D; C)$ , then $u = f + K_{q, k} u$ has a unique solution in $L^{2} (D; C)$ (Born truncation error for a small-norm volume integral operator).

[F5]

For $a, b \in L^{2} (D; C)$ , the pairing is representative-independent and $∣ \int_{D} a \overline{b} ∣ \leq ∥ a ∥_{2} ∥ b ∥_{2}$ (The complex $L^{2}$ pairing is well-defined and satisfies Cauchy–Schwarz).

[F6]

For $g \in L^{1} (R^{3}; C)$ , $\hat{g} (ξ) = \int g (y) e^{- 2 π i y \cdot ξ} d y$ is absolutely convergent at every $ξ$ and unchanged by null-set modifications (Fourier transform on complex L1 classes, The integral transform is representative independent).

[F7]

For a measurable representative with finite essential norm, $∣ q ∣ \leq ∥ q ∥_{\infty}$ almost everywhere (The essential supremum is attained as the least essential bound).

[F8]

For $v \in L^{2} (D; C)$ , the far-field map is continuous in observation direction and its values are the uniform outgoing-wave limits of the exterior volume potential, with the displayed explicit remainder (The volume far-field map is bounded and is the uniform far-field limit).

[A1]

Countable Choice is assumed by the volume model and the Neumann theorem used here (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

Given: The data in the Statement, with $q \in U_{D, k}$ .

1.1F1F2algebra

The integral operator depends linearly on $q$ , so $K_{q, k} - K_{q^{'}, k} = K_{q - q^{'}, k}$ for any $q, q^{'} \in L^{\infty} (D; C)$ . By [F2], $∥ K_{q, k} - K_{q^{'}, k} ∥ \leq C_{D} ∥ q - q^{'} ∥_{\infty}$ . The operator-norm triangle inequality then gives $∣ ρ_{k} (q) - ρ_{k} (q^{'}) ∣ \leq C_{D} ∥ q - q^{'} ∥_{\infty}$ . Thus $q \mapsto ρ_{k} (q)$ is continuous, $U_{D, k} = {ρ_{k} < 1}$ is open, and $0 \in U_{D, k}$ because $K_{0, k} = 0$ .

1.2A1F3F4given

By [F3], $f_{k, d} (x) = e^{i k d \cdot x}$ belongs to $L^{2} (D; C)$ and has norm $\sqrt{M}$ . Since $q \in U_{D, k}$ , [F4] applies with $f = f_{k, d}$ and gives exactly one $u_{q, k, d} \in L^{2} (D; C)$ .

1.3F1F3F5F7given

By [F7], $q u_{q, k, d} \in L^{2} (D)$ with norm at most $∥ q ∥_{\infty} ∥ u_{q, k, d} ∥_{2}$ , while [F3] puts $e^{i k θ \cdot y}$ in $L^{2} (D)$ . Their pairing in [F5] is the stated integral, so it is absolutely convergent and independent of the representatives. Cauchy–Schwarz gives its absolute value at most $∥ q ∥_{\infty} \sqrt{M} ∥ u_{q, k, d} ∥_{2}$ .

1.4F5F6F7givenalgebra

By [F7], $∥ q ∥_{2} \leq \sqrt{M} ∥ q ∥_{\infty}$ ; pairing $∣ q ∣$ with $1_{D}$ in [F5] then gives $\int_{D} ∣ q (y) ∣ d y \leq M ∥ q ∥_{\infty}$ . Hence $q_{0} \in L^{1} (R^{3}; C)$ . By [F6], its transform is finite and representative-independent. Since $e^{- i k θ \cdot y} e^{i k d \cdot y} = e^{- 2 π i y \cdot (k (θ - d) / (2 π))}$ , the Fourier and integral formulas for $B_{k} q$ agree and are well-defined. This proves the assertion for both data families.

2.1F8step 1.2∎

Apply [F8] to $v = u_{q, k, d}$ , whose existence is established in step 1.2. It supplies continuity in observation direction and the stated uniform outgoing-wave limit estimate, so the defined data are the physical far-field amplitude for this volume branch.

Remarks

The proof establishes only the small-norm branch selected by $ρ_{k} (q) < 1$ . It does not import global scattering uniqueness from smoother coefficient models.

Depends on

Used by

The first Born data are the Fréchet linearization at zero contrast Lemma

Cited to discharge well-definedness by Fixed-frequency far-field data on the small-norm volume branch.

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Sources

Fioralba Cakoni, David Colton, and Houssem Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS 88 (standard reference, not scraped)
Otmar Scherzer and Leonidas Mindrinos, Inverse Problems: Lecture Notes, University of Vienna, Sommersemester 2015 (standard reference, not scraped)