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Lemma 3.3: observability modulo a compact remainder

PDF p. 9: Lemma 3.3; (3.8); PDF p. 10: negative-index proof; PDF p. 11: proof conclusion

Statement

Assume the Axiom of Choice and (H): $n \geq 2$ , $Θ \subset R^{n}$ is nonempty and compact with smooth boundary, its exterior $Ω = R^{n} ∖ Θ$ is connected, and $Θ$ is non-trapping, meaning that every admissible generalized reflected ray, including glancing behavior, leaves each compact set in finite forward time. Let $H_{D}^{s} (Ω)$ be the Dirichlet spectral scale, $W (t) = e^{i t Δ_{D}}$ its unitary evolution, and $φ, ψ$ the cutoffs of Control region and nested real cutoffs.

For every $T > 0$ and $σ \in [- 2, 2)$ there is $C > 0$ such that, for every $v_{0} \in H_{D}^{σ} (Ω)$ , $∥ v_{0} ∥_{H_{D}^{σ}}^{2} \leq C (\int_{0}^{T} ∥ φ W (t) v_{0} ∥_{H_{D}^{σ}}^{2} d t + Q_{σ} (v_{0})) .$ Here $Q_{σ}$ is the lower-order observation remainder of def-se-compact-remainder: with $R = (I - Δ_{D})^{- 1}$ , $k = 0$ for $σ \geq 0$ and $k = 1$ for $σ < 0$ , and $ρ = σ + 1$ for $- 1 < σ < 0$ and $ρ = 0$ otherwise, it is $Q_{σ} (v_{0}) = \int_{0}^{T} ∥ ψ R^{k} W (t) v_{0} ∥_{H_{D}^{ρ}}^{2} d t$ .

Facts & Assumptions

[F1]

The cutoffs satisfy $χ = 1 - φ$ and $0 \leq χ \leq ψ$ ; multiplication by $φ, χ, ψ$ is bounded on the scale levels used below. Control region and nested real cutoffs Cutoff multipliers through order two and by duality

[F2]

The flow is unitary on each spectral scale level, commutes with $R$ , and $R : H_{D}^{s} \to H_{D}^{s + 2}$ is an onto isometry. Higher scale levels embed contractively into lower ones. Scale isometries and free Schrödinger evolution Positive spectral Hilbert scale: duality and interpolation The Axiom of Choice

[F3]

For $0 < τ < 2$ , the local lower-order estimate gives $\int_{0}^{T} ∥ χ W (t) u ∥_{H_{D}^{τ}}^{2} d t \leq C_{1} ∥ u ∥_{H_{D}^{θ}}^{2}$ for some $0 \leq θ < τ$ , with the explicit piecewise $θ$ in Local observation has a lower initial regularity cost. Its hypotheses are (H), Choice, and the stated cutoff and flow conventions. Local observation has a lower initial regularity cost

[F4]

For $0 \leq θ < τ$ and every $ε > 0$ , spectral absorption gives $∥ u ∥_{H_{D}^{θ}}^{2} \leq ε ∥ u ∥_{H_{D}^{τ}}^{2} + C_{ε} ∥ u ∥_{2}^{2}$ . Spectral interpolation and lower-order absorption

[F5]

For $- 2 \leq σ < 0$ , put $τ = σ + 2$ and $y = R v$ . The commutator estimate is $∥ φ y ∥_{H_{D}^{τ}} \leq ∥ φ v ∥_{H_{D}^{σ}} + C_{φ} ∥ ψ y ∥_{H_{D}^{ρ}}$ , with $ρ$ as in the Statement. Integer resolvent commutator and localized first-order bounds

Proof

[F1] The cutoffs satisfy $χ = 1 - φ$ and $0 \leq χ \leq ψ$ ; multiplication by $φ, χ, ψ$ is bounded on the scale levels used below. Control region and nested real cutoffs Cutoff multipliers through order two and by duality

[F2] The flow is unitary on each spectral scale level, commutes with $R$ , and $R : H_{D}^{s} \to H_{D}^{s + 2}$ is an onto isometry. Higher scale levels embed contractively into lower ones. Scale isometries and free Schrödinger evolution Positive spectral Hilbert scale: duality and interpolation The Axiom of Choice

[F3] For $0 < τ < 2$ , the local lower-order estimate gives $\int_{0}^{T} ∥ χ W (t) u ∥_{H_{D}^{τ}}^{2} d t \leq C_{1} ∥ u ∥_{H_{D}^{θ}}^{2}$ for some $0 \leq θ < τ$ , with the explicit piecewise $θ$ in Local observation has a lower initial regularity cost. Its hypotheses are (H), Choice, and the stated cutoff and flow conventions. Local observation has a lower initial regularity cost

[F4] For $0 \leq θ < τ$ and every $ε > 0$ , spectral absorption gives $∥ u ∥_{H_{D}^{θ}}^{2} \leq ε ∥ u ∥_{H_{D}^{τ}}^{2} + C_{ε} ∥ u ∥_{2}^{2}$ . Spectral interpolation and lower-order absorption

[F5] For $- 2 \leq σ < 0$ , put $τ = σ + 2$ and $y = R v$ . The commutator estimate is $∥ φ y ∥_{H_{D}^{τ}} \leq ∥ φ v ∥_{H_{D}^{σ}} + C_{φ} ∥ ψ y ∥_{H_{D}^{ρ}}$ , with $ρ$ as in the Statement. Integer resolvent commutator and localized first-order bounds

Given: The hypotheses, $T > 0$ , $σ \in [- 2, 2)$ , and $v_{0} \in H_{D}^{σ}$ in the Statement.

Goal: Establish the displayed estimate with a constant independent of $v_{0}$ .

1.1F1F2given

First let $0 \leq τ < 2$ , $u \in H_{D}^{τ}$ , and $z (t) = W (t) u$ . Unitarity, $z = φ z + χ z$ , and the squared triangle inequality give $T ∥ u ∥_{H_{D}^{τ}}^{2} \leq 2 \int_{0}^{T} ∥ φ z ∥_{H_{D}^{τ}}^{2} d t + 2 \int_{0}^{T} ∥ χ z ∥_{H_{D}^{τ}}^{2} d t$ . If $τ = 0$ , $0 \leq χ \leq ψ$ bounds the second integral by $\int_{0}^{T} ∥ ψ z ∥_{2}^{2} d t$ , proving the desired positive-index estimate at zero.

2.1F1F2F3F4step 1.1given

Suppose $0 < τ < 2$ . Apply the local lower-order estimate and then absorption, choosing $ε$ so that $2 C_{1} ε \leq T / 2$ . It follows that $∥ u ∥_{H_{D}^{τ}}^{2} \leq C (\int_{0}^{T} ∥ φ z ∥_{H_{D}^{τ}}^{2} d t + ∥ u ∥_{2}^{2})$ . By $L^{2}$ unitarity, $z = φ z + χ z$ , $χ \leq ψ$ , and the scale embedding into $L^{2}$ , we also have $∥ u ∥_{2}^{2} \leq (2 / T) \int_{0}^{T} (∥ φ z ∥_{H_{D}^{τ}}^{2} + ∥ ψ z ∥_{2}^{2}) d t$ . Thus, for every $0 \leq τ < 2$ , $∥ u ∥_{H_{D}^{τ}}^{2} \leq C \int_{0}^{T} (∥ φ W (t) u ∥_{H_{D}^{τ}}^{2} + ∥ ψ W (t) u ∥_{2}^{2}) d t$ .

3.1F2F5step 2.1given

Now let $- 2 \leq σ < 0$ and put $τ = σ + 2 \in [0, 2)$ , $y_{0} = R v_{0}$ , $v (t) = W (t) v_{0}$ , and $y (t) = W (t) y_{0} = R v (t)$ . The resolvent isometry gives $∥ y_{0} ∥_{H_{D}^{τ}} = ∥ v_{0} ∥_{H_{D}^{σ}}$ . Applying step 2.1 to $y_{0}$ and the commutator estimate pointwise gives $∥ v_{0} ∥_{H_{D}^{σ}}^{2} \leq C \int_{0}^{T} (∥ φ v (t) ∥_{H_{D}^{σ}}^{2} + ∥ ψ y (t) ∥_{H_{D}^{ρ}}^{2} + ∥ ψ y (t) ∥_{2}^{2}) d t$ .

4.1F1F2step 2.1step 3.1given∎

Here $ρ \geq 0$ , so the scale embedding bounds the last integrand by $∥ ψ y (t) ∥_{H_{D}^{ρ}}^{2}$ . By the definition of $Q_{σ}$ this is precisely the required estimate for negative $σ$ . For nonnegative $σ$ , step 2.1 is the required estimate because $k = 0$ and $ρ = 0$ . The flow continuity and bounded cutoff actions make the displayed time integrals well-defined.

Source notes

The article is cited for attribution of Lemma 3.3 and its proof structure on PDF pages 9–11. The deduction above uses the stated suppliers, not the article's assertion as a premise.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2