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Open questions and potential solutions

Insights, fruitful questions and potential solutions arising from Nonlinear Schrödinger equation in an exterior domain.

1. Research insights beyond the article

Research insight 01 · A quantitative regime where trajectory observability needs no separate UCP assumption

corollaryproved

Fix $σ \in [- 2, 2]$ and $T > 0$ . On the realification $H = H_{D}^{σ}$ , let $W$ be the free unitary group and $B = M_{φ}$ . Suppose $∥ B W (\cdot) x ∥_{L^{2} (0, T; H)} \geq c ∥ x ∥$ , $c > 0$ . Let $U$ solve $u_{t} = i Δ_{D} u + P (t) u$ for strongly measurable bounded real-linear $P$ with $b = \int_{0}^{T} ∥ P (t) ∥ d t < \infty$ . If $d = \sqrt{T} ∥ B ∥ (e^{b} - 1) < c$ , then $∥ B U (\cdot, 0) x ∥_{L^{2}} \geq (c - d) ∥ x ∥$ . Thus sufficiently small cubic trajectories in the stated $W^{2, \infty}$ norm satisfy energy-level UCP for this fixed time.

Why it matters

This identifies an explicit partial regime of §5.2(A) and connects exterior control to robustness of observation under bounded non-autonomous perturbations.

Originality

All 23 article pages were checked. The article develops a conditional UCP/compactness argument but does not state this quantitative small-potential criterion. Small bounded perturbation robustness is established general control theory, as noted by Haak–Hoang–Ouhabaz; no worldwide novelty is claimed.

Current knowledge

The bounded-observer inequality below is a direct derivation, consistent with the independent perturbation discussion in §5 of Haak–Hoang–Ouhabaz. It is a report consequence, without library publication or an independent acceptance stamp.

Argument and next steps

Duhamel and Gronwall give $∥ U (t, 0) ∥ \leq \exp (\int_{0}^{t} ∥ P (r) ∥ d r)$ . Subtracting $W (t)$ and integrating $∥ P (r) ∥ \exp (\int_{0}^{r} ∥ P ∥)$ gives $∥ U (t, 0) - W (t) ∥ \leq e^{b} - 1$ uniformly. Hence $∥ B (U - W) x ∥_{L^{2}} \leq \sqrt{T} ∥ B ∥ (e^{b} - 1) ∥ x ∥$ ; the reverse triangle inequality proves the claim. For the cubic adjoint, $P (t) v = i V_{1} v - i V_{2} \overline{v}$ . At $σ = 1$ , its norm is at most $C ∣ λ ∣ ∥ w (t) ∥_{W^{2, \infty}}^{2}$ . Sufficiently small $\int_{0}^{T} ∥ w (t) ∥_{W^{2, \infty}}^{2} d t$ therefore makes $d < c$ . Energy-level vanishing on $ω$ implies $B v = 0$ , so the positive lower bound forces zero initial data and uniqueness gives $v = 0$ .

Remaining gaps

The threshold depends on the free observability constant, the cutoff and T; it is not uniform as T tends to zero and does not treat arbitrary large trajectories. For σ≠0 the multiplier norm ||B|| need not equal one. Real-linearity is preserved throughout; no physical postulate is a mathematical premise.

Sources and reading limits · 2

Nonlinear Schrödinger equation in an exterior domain, frozen v2: https://arxiv.org/abs/2610.11968; PDF pp.1–23, all statements, proofs, figures, footnotes and bibliography; Proposition 3.5 and Theorem 3.6, pp.11–13; Theorem 3.7 and Propositions 3.9–3.10, pp.14–16; §4, pp.17–21; §5.2(A)–(F), pp.21–22. Reading limits: Read all 23 frozen PDF pages in text and rendered images, and all 118 original-item records. Verified PDF SHA256 baa8dd08f7d5a4f2e96a06fd0b78f448a7b922224dd11211ddb41de4aa44eb9d. This source supplies attribution and starting theorems for report consequences; it is not evidence for its own prerequisites. The live arXiv search metadata was not substituted for the frozen PDF.
Haak, Hoang and Ouhabaz, Controllability and observability for non-autonomous evolution equations: the averaged Hautus test: https://www.math.u-bordeaux.fr/~bhaak/recherche/HHO.pdf; PDF p.2, introduction, footnote on small bounded perturbations; §5, PDF pp.11–12, equations (5.1)–(5.5) and following discussion. Reading limits: Read the introduction and the §5 perturbation/evolution discussion in extracted text, including its hypotheses and Duhamel formula. The screenshot request for p.11 failed; no rendered-formula inspection is claimed. The report proves its simpler bounded-observer estimate directly; it does not import the paper’s more general averaged Hautus theorem or its bounded-domain application.

Research insight 02 · Additive control has a mass budget distinct from unitary state steering

connectionproved

For $n = 2, 3$ , $m \in {3, 5}$ , a solution $v \in C ([0, T]; H_{D}^{2})$ with continuous $H_{D}^{2}$ control satisfies $M^{'} (t) = 2 Im \int_{Ω} φ h \overline{v} d x$ , where $M = ∥ v ∥_{2}^{2}$ . Consequently $∣ ∥ v (T) ∥_{2} - ∥ v (0) ∥_{2} ∣ \leq \int_{0}^{T} ∥ φ h (t) ∥_{2} d t \leq \sqrt{T} ∥ φ h ∥_{L_{t}^{2} L_{x}^{2}}$ . This is a necessary forcing budget for unequal-mass endpoints. Real multiplicative forcing $a v$ has zero mass contribution, under regularity permitting the same calculation.

Why it matters

The article names quantum and condensate applications but uses complex additive forcing. The identity clarifies which mathematical control model allows unequal-norm endpoints.

Originality

The whole article was checked: it does not derive this endpoint mass-budget bound or compare additive forcing with norm-preserving potential control. The conservation-law calculation and unitary norm preservation are standard; no worldwide novelty is asserted.

Current knowledge

Proved below at the order-two regularity actually available in the paper. Teschl’s self-adjoint evolution theorem independently supplies the unitary comparison. The Physics Library’s flow lemma has domain mathematics; its isolated-dynamics postulate has domain physics and is not used as a PDE premise.

Argument and next steps

The exterior $H^{2}$ algebra for $n = 2, 3$ makes the polynomial nonlinearity continuous in $L^{2}$ ; the equation therefore gives $v_{t} \in C_{t} L^{2}$ . Differentiate $∥ v ∥_{2}^{2}$ . Dirichlet integration by parts makes $\int Δ v \overline{v} = - ∥ \nabla v ∥_{2}^{2}$ real, and $λ \int ∣ v ∣^{m + 1}$ is real. The remaining term is $2 Re (- i \int φ h \overline{v}) = 2 Im \int φ h \overline{v}$ . Cauchy–Schwarz gives $∣ M^{'} ∣ \leq 2 ∥ φ h ∥_{2} \sqrt{M}$ . Apply the resulting bound to $\sqrt{M + ε}$ , integrate and let $ε ↓ 0$ , then use time Cauchy–Schwarz. For forcing $a v$ with real $a$ , $\int a ∣ v ∣^{2}$ is real, so the mass contribution vanishes.

Remaining gaps

Only the stated sufficiently regular solutions are covered by this derivation; no unproved extension to arbitrary distributional solutions is claimed. The model comparison does not identify an experimental actuator, a microscopic condensate derivation or measured mass transfer.

Sources and reading limits · 2

Nonlinear Schrödinger equation in an exterior domain, frozen v2: https://arxiv.org/abs/2610.11968; PDF pp.1–23, all statements, proofs, figures, footnotes and bibliography; Proposition 3.5 and Theorem 3.6, pp.11–13; Theorem 3.7 and Propositions 3.9–3.10, pp.14–16; §4, pp.17–21; §5.2(A)–(F), pp.21–22. Reading limits: Read all 23 frozen PDF pages in text and rendered images, and all 118 original-item records. Verified PDF SHA256 baa8dd08f7d5a4f2e96a06fd0b78f448a7b922224dd11211ddb41de4aa44eb9d. This source supplies attribution and starting theorems for report consequences; it is not evidence for its own prerequisites. The live arXiv search metadata was not substituted for the frozen PDF.
Gerald Teschl, Mathematical Methods in Quantum Mechanics, online 2009 edition: https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf; §5.1, Theorem 5.1 and proof, printed pp.123–124, PDF pp.135–136; Theorem 5.2, printed pp.124–125, PDF pp.136–137. Reading limits: Read the stated self-adjoint/unitary generation theorem and its proof, and Stone’s theorem in extracted text. Used only for mathematical comparison with norm-preserving unitary evolution. No experiment, condensate approximation or exterior-control apparatus is inferred.

2. Potentially fruitful questions

Research direction 01

For every smooth nonempty compact non-trapping obstacle with connected exterior, $T > 0$ , nonzero real $λ$ , and cubic trajectory $w$ satisfying (W), does every $v \in C ([0, T]; H_{0}^{1} (Ω))$ solving $i v_{t} + Δ v + 2 λ ∣ w ∣^{2} v - λ w^{2} \overline{v} = 0$ and vanishing on $ω \times (0, T)$ vanish identically?

Why it matters

An affirmative answer would remove the explicit UCP hypothesis in the trajectory-control theorems.

Originality

This is the authors’ §5.2(A), not a new question. Full-article reading checked that Lemma 3.4 only settles the free case and Lemma 4.5 only regularizes invisible solutions. The small-potential criterion above gives a restricted consequence, not worldwide novelty.

Current knowledge

The frozen October 2026 article expressly leaves the general case open. Targeted searches on 13 November 2026 did not locate a later primary resolution with these exact exterior and conjugate-coupled hypotheses. The elliptic three-dimensional library UCP theorem does not settle this time-dependent problem.

Potential solution

Partial solutionProved partial result

The energy-level implication holds when $\sqrt{T} ∥ B ∥ (e^{b_{1}} - 1) < c_{1}$ , where $b_{1} = \int_{0}^{T} ∥ i M_{V_{1}} - i M_{V_{2}} K ∥_{H_{D}^{1} \to H_{D}^{1}} d t$ , $K$ is conjugation and $c_{1}$ is the free observation lower bound.

Argument and next steps

The bounded-perturbation derivation gives $∥ B v ∥_{L_{t}^{2} H_{D}^{1}} \geq (c_{1} - \sqrt{T} ∥ B ∥ (e^{b_{1}} - 1)) ∥ v (0) ∥_{H_{D}^{1}}$ . Vanishing on $ω$ gives $B v = 0$ . The positive lower bound forces zero initial data, hence zero evolution by uniqueness.

Remaining gaps

No conclusion for arbitrary trajectories or large b₁. This does not prove the missing general boundary Carleman estimate.

Sources and reading limits · 2

Nonlinear Schrödinger equation in an exterior domain, frozen v2: https://arxiv.org/abs/2610.11968; PDF pp.1–23, all statements, proofs, figures, footnotes and bibliography; Proposition 3.5 and Theorem 3.6, pp.11–13; Theorem 3.7 and Propositions 3.9–3.10, pp.14–16; §4, pp.17–21; §5.2(A)–(F), pp.21–22. Reading limits: Read all 23 frozen PDF pages in text and rendered images, and all 118 original-item records. Verified PDF SHA256 baa8dd08f7d5a4f2e96a06fd0b78f448a7b922224dd11211ddb41de4aa44eb9d. This source supplies attribution and starting theorems for report consequences; it is not evidence for its own prerequisites. The live arXiv search metadata was not substituted for the frozen PDF.
Haak, Hoang and Ouhabaz, Controllability and observability for non-autonomous evolution equations: the averaged Hautus test: https://www.math.u-bordeaux.fr/~bhaak/recherche/HHO.pdf; PDF p.2, introduction, footnote on small bounded perturbations; §5, PDF pp.11–12, equations (5.1)–(5.5) and following discussion. Reading limits: Read the introduction and the §5 perturbation/evolution discussion in extracted text, including its hypotheses and Duhamel formula. The screenshot request for p.11 failed; no rendered-formula inspection is claimed. The report proves its simpler bounded-observer estimate directly; it does not import the paper’s more general averaged Hautus theorem or its bounded-domain application.

Research direction 02

For fixed admissible non-trapping exterior $Ω$ , cutoff $φ$ and $σ \in [- 2, 2]$ , determine the small-time growth of $C_{σ} (T) = \sup_{x \neq 0} ∥ x ∥_{H_{D}^{σ}} / ∥ φ e^{i t Δ_{D}} x ∥_{L^{2} (0, T; H_{D}^{σ})}$ . When the uncontrolled region has nonempty interior, do positive finite $a, A$ give $e^{a / T} ≲ C_{σ} (T) ≲ e^{A / T}$ for small $T$ , and which geometric quantities govern them?

Why it matters

Existence for every positive time does not quantify forcing cost, the small-trajectory robustness threshold, or the nonlinear controllable neighborhood.

Originality

The article cites fast-control literature but does not formulate or estimate this exterior spectral-scale cost. Miller already proves exponential geometric bounds for compact manifolds under his stated hypotheses; the exterior formulation is a proposed extension, not a new general phenomenon.

Current knowledge

Read Miller’s Definition 1, Theorems 1.1 and 1.3 and the localized construction in §2.1. Those compact-manifold statements do not supply the requested unbounded-domain conclusion. The targeted search did not establish a current complete result for this exact cutoff/scale formulation; its worldwide novelty and status remain unverified.

Potential solution

Partial solutionProved partial result

The elementary lower bound $C_{σ} (T) \geq 1 / (\sqrt{T} ∥ M_{φ} ∥_{H_{D}^{σ} \to H_{D}^{σ}})$ holds.

Argument and next steps

Unitarity gives $∥ φ W (t) x ∥_{H_{D}^{σ}} \leq ∥ M_{φ} ∥ ∥ x ∥_{H_{D}^{σ}}$ . Its time-integrated norm is at most $\sqrt{T} ∥ M_{φ} ∥ ∥ x ∥$ . Taking the supremum of the reciprocal ratio proves the bound. Exponential bounds require additional localized-wavepacket and quantitative-observability estimates.

Remaining gaps

This polynomial lower bound does not establish either exponential estimate or its sharp geometric constants.

Sources and reading limits · 2

Nonlinear Schrödinger equation in an exterior domain, frozen v2: https://arxiv.org/abs/2610.11968; PDF pp.1–23, all statements, proofs, figures, footnotes and bibliography; Proposition 3.5 and Theorem 3.6, pp.11–13; Theorem 3.7 and Propositions 3.9–3.10, pp.14–16; §4, pp.17–21; §5.2(A)–(F), pp.21–22. Reading limits: Read all 23 frozen PDF pages in text and rendered images, and all 118 original-item records. Verified PDF SHA256 baa8dd08f7d5a4f2e96a06fd0b78f448a7b922224dd11211ddb41de4aa44eb9d. This source supplies attribution and starting theorems for report consequences; it is not evidence for its own prerequisites. The live arXiv search metadata was not substituted for the frozen PDF.
Luc Miller, How violent are fast controls for Schrödinger and plate vibrations?, arXiv:math/0309040v3: https://arxiv.org/pdf/math/0309040; §1.1, PDF p.2, Definition 1 and compact-manifold hypotheses; §1.2, PDF pp.2–3, Theorems 1.1 and 1.3; §2.1, PDF pp.6–8, Theorem 2.1 and localized heat-kernel construction. Reading limits: Read these statements and the displayed localization argument in the PDF text extraction. The compact-manifold and generalized-geodesic hypotheses remain attached. No direct application of those theorems to an unbounded obstacle exterior is asserted; the exponential-rate analogy motivates an unresolved question.

Research direction 03

For the defocusing quintic equation $i v_{t} + Δ v - ∣ v ∣^{4} v = φ h$ on a smooth non-trapping exterior in $R^{3}$ with homogeneous Dirichlet data, can every pair in $H_{0}^{1} (Ω)$ be connected by an energy-class solution in a sufficiently large finite time, using $h \in C_{t} H_{0}^{1}$ acting through $φ$ outside a ball?

Why it matters

This seeks a large-data analogue of the small-data critical corollary; fixed-point smallness alone cannot yield it.

Originality

This specializes the large-data direction in §5.2(C) to a precise defocusing internal-control problem. It is author-origin, with a related whole-space question in the independent critical-case paper, §4(B); no worldwide novelty is claimed.

Current knowledge

The frozen article leaves it open. The independently read critical-case v3, Theorem 1.1, proves a small-data whole-space null-control statement and §4(A)–(B) still asks about stabilization and large-data control. A targeted search did not establish a later primary resolution for the stated exterior problem.

Potential solution

No solution establishedUnresolved

No large-data exterior controllability theorem is derived here.

Argument and next steps

Both primary sources suggest stabilization followed by local control. A proof would need a global controlled/damped energy-class evolution and decay reaching the local-control ball in the same norm and geometry. Neither the small-data corollary nor the available L² damping comparison supplies these steps in the critical H¹ setting.

Remaining gaps

No matching critical stabilization theorem was established in this assessment; the proposed route remains incomplete.

Sources and reading limits · 2

Nonlinear Schrödinger equation in an exterior domain, frozen v2: https://arxiv.org/abs/2610.11968; PDF pp.1–23, all statements, proofs, figures, footnotes and bibliography; Proposition 3.5 and Theorem 3.6, pp.11–13; Theorem 3.7 and Propositions 3.9–3.10, pp.14–16; §4, pp.17–21; §5.2(A)–(F), pp.21–22. Reading limits: Read all 23 frozen PDF pages in text and rendered images, and all 118 original-item records. Verified PDF SHA256 baa8dd08f7d5a4f2e96a06fd0b78f448a7b922224dd11211ddb41de4aa44eb9d. This source supplies attribution and starting theorems for report consequences; it is not evidence for its own prerequisites. The live arXiv search metadata was not substituted for the frozen PDF.
Braz e Silva, Capistrano–Filho, Carvalho and Dos Santos Ferreira, Control of the Schrödinger equation in R3: The critical case, v3: https://arxiv.org/html/2404.07749v3; Theorem 1.1, local small-H1 null controllability; §4, Comments and open issues, items (A) stabilization and (B) global control. Reading limits: Read the arXiv v3 theorem statement and §4 in HTML; the abstract page identifies revision 12 April 2026. The proof was not audited. This is evidence for the authors’ stated local theorem and listed large-data direction, not a comprehensive 2026 worldwide status certificate.

3. Potential solutions to open questions

Research direction 01

Question source

Origin: Original authors

https://arxiv.org/abs/2610.11968; Frozen v2, PDF p.21, §5.2(A)

Why it matters

An affirmative answer would remove the explicit UCP hypothesis in the trajectory-control theorems.

Originality

A restricted consequence of standard bounded-perturbation robustness, absent from the article’s stated results. It partially addresses the authors’ question without claiming a novel general UCP theorem.

Current knowledge

The independent non-autonomous observability paper notes small bounded perturbation robustness. The fixed-time estimate is derived explicitly in insight-se-quantitative-small-perturbation; general trajectories remain outside its scope.

Potential solution

Partial solutionProved partial result

Argument and next steps

Remaining gaps

No conclusion for arbitrary trajectories or large b₁. This does not prove the missing general boundary Carleman estimate.

Sources and reading limits · 2

Nonlinear Schrödinger equation in an exterior domain, frozen v2: https://arxiv.org/abs/2610.11968; PDF pp.1–23, all statements, proofs, figures, footnotes and bibliography; Proposition 3.5 and Theorem 3.6, pp.11–13; Theorem 3.7 and Propositions 3.9–3.10, pp.14–16; §4, pp.17–21; §5.2(A)–(F), pp.21–22. Reading limits: Read all 23 frozen PDF pages in text and rendered images, and all 118 original-item records. Verified PDF SHA256 baa8dd08f7d5a4f2e96a06fd0b78f448a7b922224dd11211ddb41de4aa44eb9d. This source supplies attribution and starting theorems for report consequences; it is not evidence for its own prerequisites. The live arXiv search metadata was not substituted for the frozen PDF.
Haak, Hoang and Ouhabaz, Controllability and observability for non-autonomous evolution equations: the averaged Hautus test: https://www.math.u-bordeaux.fr/~bhaak/recherche/HHO.pdf; PDF p.2, introduction, footnote on small bounded perturbations; §5, PDF pp.11–12, equations (5.1)–(5.5) and following discussion. Reading limits: Read the introduction and the §5 perturbation/evolution discussion in extracted text, including its hypotheses and Duhamel formula. The screenshot request for p.11 failed; no rendered-formula inspection is claimed. The report proves its simpler bounded-observer estimate directly; it does not import the paper’s more general averaged Hautus theorem or its bounded-domain application.

Research direction 02

Do analogues of Theorems 1.4–1.5 hold for the quintic equation near $w \in C_{t} (H_{D}^{2} \cap W^{2, \infty})$ , assuming its corresponding energy-level unique-continuation property?

Question source

Origin: Original authors

https://arxiv.org/abs/2610.11968; Frozen v2, PDF p.21, §5.2(C), first two sentences

Why it matters

The authors suggest this conditional extension but do not supply its polynomial remainder estimates or a theorem.

Originality

The direction and linearized coefficients are explicitly the authors’. The derivation below fills in that conditional composition; it is not an insight absent from the article’s discussion, a general UCP solution, or a claim of worldwide priority.

Current knowledge

Full-article reading checked that only the zero-trajectory quintic result is proved there. Targeted searches located no separate primary theorem resolving this precise conditional formulation; the proof below is a report derivation from the article’s applicable linear estimates, with its added quintic UCP assumption explicitly retained.

Potential solution

Full solutionProved solution

Assume (H), $n = 2, 3$ , $T > 0$ , nonzero real $λ$ , and a quintic trajectory $w \in C ([0, T]; H_{D}^{2} \cap W^{2, \infty})$ with zero Dirichlet trace. Assume energy-level UCP for $i v_{t} + Δ v + 3 λ ∣ w ∣^{4} v - 2 λ ∣ w ∣^{2} w^{2} \overline{v} = 0$ . For $s = 1, 2$ , the quintic linearization is exactly controllable with continuous $H_{D}^{s}$ controls, and the nonlinear quintic equation is locally exactly controllable in $H_{D}^{s}$ about the endpoints of $w$ .

Argument and next steps

Set $g (u) = λ u^{3} {\overline{u}}^{2}$ . Its real derivative is $D g (w) z = V_{1} z + V_{2} \overline{z}$ , with $V_{1} = 3 λ ∣ w ∣^{4}$ and $V_{2} = 2 λ ∣ w ∣^{2} w^{2}$ . Both are continuous $W^{2, \infty}$ multipliers. Lemmas 4.1–4.5 use only these coefficient bounds, the real adjoint signs and free estimates, not the cubic trajectory equation. Thus the same regularization, compactness and invisible-solution bootstrap apply. Quintic UCP removes the compact remainder, and the real HUM calculation gives a bounded continuous-control map $G_{5}$ . The remainder is $F_{5} (w, z) = λ \sum_{a + b \geq 2} (\binom{3}{a}) (\binom{2}{b}) w^{3 - a} {\overline{w}}^{2 - b} z^{a} {\overline{z}}^{b}$ , with $0 \leq a \leq 3$ , $0 \leq b \leq 2$ . Its degrees in $z$ are $j = 2, \dots, 5$ . For $s = 1$ , coefficients have bounded $W^{1, \infty}$ norms. The product rule bounds each degree by $C_{w} ∥ z ∥_{\infty}^{j - 1} ∥ z ∥_{H^{1}}$ . In $X = C_{t} H_{0}^{1} \cap L_{t}^{4} L_{x}^{\infty}$ , time Hölder gives $∥ F_{5} (w, z) ∥_{L_{t}^{1} H^{1}} \leq C_{w} \sum_{j = 2}^{5} T^{(5 - j) / 4} ∥ z ∥_{X}^{j}$ . Expanding and differentiating a difference gives the Lipschitz bound $C_{w} \sum_{j = 2}^{5} T^{(5 - j) / 4} (∥ z ∥_{X}^{j - 1} + ∥ \tilde{z} ∥_{X}^{j - 1}) ∥ z - \tilde{z} ∥_{X}$ . Each monomial has zero trace by the bounded polynomial product argument of Proposition 3.9. For $s = 2$ , coefficient multiplication in $W^{2, \infty}$ and the exterior $H^{2}$ algebra give the same power/difference bounds with factor $T$ , preserving zero trace. Duhamel and Gronwall transfer the free homogeneous and forcing bounds to the real-linear propagator $S_{5}$ . Put $N_{5} (z) (t) = \int_{0}^{t} S_{5} (t, r) (i F_{5} (w, z) (r)) d r$ and $h_{z} = G_{5} (z_{0}, z_{T} - N_{5} (z) (T))$ . The map $S_{5} (t, 0) z_{0} + N_{5} (z) (t) + \int_{0}^{t} S_{5} (t, r) (- i φ h_{z} (r)) d r$ has the prescribed endpoints. Its norm is at most $C_{1} (∥ z_{0} ∥ + ∥ z_{T} ∥) + C_{2} \sum_{j = 2}^{5} ∥ z ∥_{X}^{j}$ , and its Lipschitz constant on a radius- $M$ ball is at most $2 C_{2} \sum_{j = 2}^{5} M^{j - 1}$ , absorbing fixed-time factors into $C_{2}$ . Choose $M$ small so this is at most $1 / 2$ and the nonlinear self-map contribution is at most $M / 2$ ; take the endpoint sum at most $M / (2 C_{1})$ . The nonempty closed ball is complete and invariant, so Banach’s theorem supplies the controlled solution. The factors $\pm i$ stay inside $S_{5}$ because it is only real-linear.

Remaining gaps

This fully resolves only the stated conditional quintic analogue. It does not establish quintic UCP for arbitrary trajectories, lower trajectory regularity, large-data control, Neumann control or a boundary trajectory theorem. It is a local derivation in this report and carries no independent review/publication stamp.

Sources and reading limits · 1

Nonlinear Schrödinger equation in an exterior domain, frozen v2: https://arxiv.org/abs/2610.11968; PDF pp.1–23, all statements, proofs, figures, footnotes and bibliography; Proposition 3.5 and Theorem 3.6, pp.11–13; Theorem 3.7 and Propositions 3.9–3.10, pp.14–16; §4, pp.17–21; §5.2(A)–(F), pp.21–22. Reading limits: Read all 23 frozen PDF pages in text and rendered images, and all 118 original-item records. Verified PDF SHA256 baa8dd08f7d5a4f2e96a06fd0b78f448a7b922224dd11211ddb41de4aa44eb9d. This source supplies attribution and starting theorems for report consequences; it is not evidence for its own prerequisites. The live arXiv search metadata was not substituted for the frozen PDF.

Review limits

Full-article novelty checks used all 23 PDF pages, including §5.2(A)–(F), rather than generated open-problem statements. Absence of the two derived insights from this article is not a claim of worldwide novelty.
Current-status searches on 13 November 2026 used the article title/identifier, exterior conjugate-coupled UCP, quintic trajectory control, bounded-perturbation observability and fast-control cost queries. Primary passages were read in Haak–Hoang–Ouhabaz, Miller, the critical-case v3 and Teschl. Search returns from AI summaries and secondary question aggregators were not used as status evidence.
No exhaustive search of all 2026 preprints, non-English literature or inaccessible sources was performed. An absence of a located resolution is reported as a search limit, not proof that a question remains globally open.
The small-potential solution is quantitative only in terms of an unknown free observation constant. The cost question has only a polynomial lower bound here, and the critical large-data question has no derived solution.
The quintic argument solves the conditional formulation suggested by the authors. Its corresponding UCP is an assumption, not a result, and the derived report consequence supplies no logical dependency to portal items.