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Real pairing cancellation for the linearized-adjoint pair

PDF p. 19: Proof of Theorem 1.4, pairing cancellation

Statement

Assume the Axiom of Choice and the geometry, trajectory $w$ , real cutoff $φ$ , coefficients $V_{1} = 2 λ ∣ w ∣^{2}$ and $V_{2} = λ w^{2}$ , and linearized and adjoint equations specified in the linearized-pair definition. Write $Δ_{D}$ for the Dirichlet Laplacian on $L^{2} (Ω)$ , and $(f, g)_{R} = Re \int_{Ω} f \overline{g} d x$ .

Let $z, v \in C^{1} ((0, T); L^{2} (Ω)) \cap C ((0, T); D (Δ_{D}))$ , where the latter space has the graph norm $∥ u ∥_{D (Δ_{D})} = (∥ u ∥_{2}^{2} + ∥ Δ_{D} u ∥_{2}^{2})^{1 / 2}$ . Suppose that $h (t) \in L^{2} (Ω)$ and that, for every $t \in (0, T)$ , the two equations hold in $L^{2} (Ω)$ with control $h (t)$ . Then

$\frac{d}{d t} (z (t), v (t))_{R} = (- i φ h (t), v (t))_{R} (0 < t < T) .$

In particular the Laplacian and both potential contributions cancel. The displayed minus sign on the conjugate-potential term in the adjoint equation supplies the sign needed for this cancellation.

Statement references

Linearized equation and its real-duality adjoint

Dirichlet spectral Sobolev scale and duality

Dirichlet Laplacian and positive shift

The Axiom of Choice

Facts & Assumptions

[F1]

The systems in Linearized equation and its real-duality adjoint have $i z_{t} + Δ_{D} z + V_{1} z + V_{2} \overline{z} = φ h$ and $i v_{t} + Δ_{D} v + V_{1} v - V_{2} \overline{v} = 0$ . The same definition supplies real $V_{1}$ , essentially bounded $V_{1}, V_{2}$ at each time, and the stated real pairing.

[F2]

Under the stated Choice assumption The Axiom of Choice, Dirichlet Laplacian and positive shift supplies the self-adjoint Dirichlet realization on $D (Δ_{D}) = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ ; Dirichlet spectral Sobolev scale and duality fixes the first-variable-linear $L^{2}$ pairing. Hence $⟨ Δ_{D} f, g ⟩ = ⟨ f, Δ_{D} g ⟩$ for $f, g \in D (Δ_{D})$ .

[F3]

The assumed full Axiom of Choice implies Countable Choice by The Axiom of Choice implies countable choice, so the Countable Choice hypothesis of Realification of Hilbert pairings and conjugate-linear coefficients is satisfied. For real essentially bounded $a$ and complex essentially bounded $b$ , Realification of Hilbert pairings and conjugate-linear coefficients gives the real adjoint of $L : f \mapsto i (a f + b \overline{f})$ as $L_{R}^{*} g = - i a g + i b \overline{g}$ , so $(L f, g)_{R} + (f, - L_{R}^{*} g)_{R} = 0$ .

[F4]

The inner-product continuity estimate of The inner product is jointly continuous implies continuity of the $L^{2}$ pairing under norm convergence in each argument.

Proof

[F1] The systems in Linearized equation and its real-duality adjoint have $i z_{t} + Δ_{D} z + V_{1} z + V_{2} \overline{z} = φ h$ and $i v_{t} + Δ_{D} v + V_{1} v - V_{2} \overline{v} = 0$ . The same definition supplies real $V_{1}$ , essentially bounded $V_{1}, V_{2}$ at each time, and the stated real pairing.

[F2] Under the stated Choice assumption The Axiom of Choice, Dirichlet Laplacian and positive shift supplies the self-adjoint Dirichlet realization on $D (Δ_{D}) = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ ; Dirichlet spectral Sobolev scale and duality fixes the first-variable-linear $L^{2}$ pairing. Hence $⟨ Δ_{D} f, g ⟩ = ⟨ f, Δ_{D} g ⟩$ for $f, g \in D (Δ_{D})$ .

[F3] The assumed full Axiom of Choice implies Countable Choice by The Axiom of Choice implies countable choice, so the Countable Choice hypothesis of Realification of Hilbert pairings and conjugate-linear coefficients is satisfied. For real essentially bounded $a$ and complex essentially bounded $b$ , Realification of Hilbert pairings and conjugate-linear coefficients gives the real adjoint of $L : f \mapsto i (a f + b \overline{f})$ as $L_{R}^{*} g = - i a g + i b \overline{g}$ , so $(L f, g)_{R} + (f, - L_{R}^{*} g)_{R} = 0$ .

[F4] The inner-product continuity estimate of The inner product is jointly continuous implies continuity of the $L^{2}$ pairing under norm convergence in each argument.

Given: The stated Choice assumption, geometry, trajectory, coefficients, control, and strong solutions $z, v$ satisfying the two equations in $L^{2}$ at every time.

Goal: Prove the displayed derivative identity for each $t \in (0, T)$ .

1.1F1F3algebra

Fix $t \in (0, T)$ . The equations give $z_{t} = i Δ_{D} z + i (V_{1} z + V_{2} \overline{z}) - i φ h$ and $v_{t} = i Δ_{D} v + i V_{1} v - i V_{2} \overline{v}$ in $L^{2}$ , all evaluated at $t$ . Set $L_{t} f = i (V_{1} f + V_{2} \overline{f})$ ; then the potential term in $v_{t}$ is $- L_{t, R}^{*} v$ . The boundedness and reality hypotheses needed for this real adjoint hold at the fixed time. The Choice implication in F3 supplies the Countable Choice hypothesis required for this application of the dualization lemma.

1.2F4given

The difference quotient of the pairing is $(\frac{z (t + s) - z (t)}{s}, v (t + s))_{R} + (z (t), \frac{v (t + s) - v (t)}{s})_{R}$ . The difference quotients converge in $L^{2}$ to $z_{t} (t)$ and $v_{t} (t)$ , and $v (t + s) \to v (t)$ in $L^{2}$ ; pairing continuity therefore gives $\frac{d}{d t} (z, v)_{R} = (z_{t}, v)_{R} + (z, v_{t})_{R}$ .

1.3F2algebra

Self-adjointness and the first-variable-linear convention give $⟨ i Δ_{D} z, v ⟩ = i ⟨ Δ_{D} z, v ⟩$ and $⟨ z, i Δ_{D} v ⟩ = - i ⟨ z, Δ_{D} v ⟩$ ; their sum is zero. Thus the two Laplacian terms have zero sum also after taking real parts.

2.1F3step 1.1step 1.2step 1.3∎

The two potential terms sum to $(L_{t} z, v)_{R} + (z, - L_{t, R}^{*} v)_{R} = 0$ . Substituting the equations into the product rule and using the Laplacian cancellation leaves exactly $(- i φ h, v)_{R}$ . Since $t$ was arbitrary, the identity holds throughout $(0, T)$ .

Source notes

The frozen article is cited for attribution of the smooth-data identity in the proof of Theorem 1.4. The argument above uses the independent mathematical suppliers and proves the identity in the stated strong $L^{2}$ class. It does not assert an extension to negative-order pivot pairings or establish existence of solutions.

Depends on

Used by · 1

Coercivity and bounded trajectory control operator

Dependency tree · 2 levels

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF page 19, proof of Theorem 1.4