{"id":"807eae83-9d04-4ca2-989c-385bb7de70a4","arxiv_id":"2608.04357","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Schrödinger equations on tori with real-part controls, the maximal null-controllable subspace is the real orthogonal complement of the purely imaginary stationary modes.","lead":"This paper finds exactly which initial states of a Schrödinger equation on a torus can be driven to zero when the control enters only through its real part. The answer is simple: all states except the purely imaginary stationary modes, for open regions and for a class of measurable product control regions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 rests on Proposition 4.4(iii), an unproved multiplier approximation imported from Burq–Zhu preprints; if that approximation is not exactly what their Theorem 1.10 provides, the double-spectrum argument in Lemma 4.5 collapses.","rationale":"I read the paper in full. The open-set branch leading to Theorem 1.1 is coherent: the real-Hilbert duality, the compactness-uniqueness framework, the high-frequency real-part estimate, and the unique-continuation identification N_T = K_Im all check out. I did not find the conjugate-identity error alleged in Eq. (3.9): with A = e^{itH}phi, Re A = (A + \\bar A)/2, so the displayed separation into 1/2 ||A||^2 and 1/4 (A^2 + \\bar A^2) is correct. The measurable-product branch is different. Proposition 4.4(iii) is the single external, unproved input on which Lemma 4.5, Theorem 4.3, and hence Theorem 1.2 rest. The paper cites two preprints and sketches a multiplier-norm closure argument, but it does not state the hypotheses or prove the approximation. This is a genuine load-bearing concern, not a stylistic one: without (4.7)-(4.9), the high-frequency estimate has no mechanism to replace the rough indicator by a smooth multiplier. The reader identified this same external assumption as the weakest point, so I partially agree; I do not share the reader's conjugate-error concern. The appropriate disposition remains CONDITIONAL: Theorem 1.1 appears solid, while Theorem 1.2 should be accepted only after Proposition 4.4(iii) is verified against the cited Burq-Zhu results.","tokens_in":33232,"tokens_out":30988,"duration_ms":314038,"concrete_test":"Obtain arXiv:2509.23965 and locate the theorem cited as Theorem 1.10. Check whether it states, for every admissible measurable product set G and every epsilon>0, the existence of a real trigonometric polynomial zeta with ||(1_G - zeta) e^{itDelta} f||_{L^2(Q)} <= epsilon ||e^{itDelta} f||_{L^2(Q)} for every f in L^2(T^d). If it does, verify that the proof also yields the same zeta for e^{-itDelta} and for the shifted phases in (4.9), as claimed. If the cited result is weaker, for instance observability only or approximation in a different norm, then Lemma 4.5 Step 1 fails and Theorem 1.2 requires a new argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 proves Theorem 1.2 through Theorem 4.3, the full-frequency double-spectrum inequality. Lemma 4.5 is the first load-bearing step: it separates the forward and backward branches and, in Step 1, uses Proposition 4.4(iii) to replace the indicator 1_G by a real trigonometric polynomial zeta with error epsilon in the multiplier norm on the Schrödinger solution spaces, including both e^{±itDelta} and the shifted phases e^{ict}e^{itDelta} and e^{-ict}e^{-itDelta}. Proposition 4.4(iii) is not proved in this paper: the proof in Section 4.1 simply asserts that Burq–Zhu's Theorem 1.10, together with the mixed-norm estimate (ii), places 1_G in the multiplier-norm closure of trigonometric polynomials. The exact statement of the cited theorem, its hypotheses on admissible measurable product sets, and the passage from forward to backward and shifted estimates are not reproduced. Since Lemma 4.5, Lemma 4.9, and Theorem 4.3 all inherit this approximation, Theorem 1.2 has no self-contained proof of its key input. If Proposition 4.4(iii) is not exactly what the cited preprints establish, the measurable-product characterization is unsupported. In contrast, Theorem 1.1's open-set proof does not use this input; I also checked Eq. (3.9) and the expansion there is correct, so the alleged conjugate-identity defect is not a substantive issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the null-controllable subspace of the Schrödinger equation on the torus when the control enters only through its real part. Treating L^2 as a real Hilbert space, it proves that the maximal controllable subspace is the real orthogonal complement of K_Im = {i Im φ : φ ∈ ker(-Δ+V)}: Theorem 1.1 for cylindrical open control sets G=(0,T)×E, and Theorem 1.2 for admissible measurable product sets in the shifted free case V≡-c. The proof reduces controllability to real-part observability inequalities (Propositions 2.1 and 2.2), proves the open-set estimate by a compactness-uniqueness argument (Section 3), and proves the measurable-product estimate through a double-spectrum inequality (Section 4), using Burq-Zhu's complex-valued observability and multiplier approximation results as external inputs. The paper also establishes optimality of the stationary correction term and includes a coercivity lemma for nonnegative potentials in the appendix.","tokens_in":33533,"tokens_out":11708,"duration_ms":116205,"significance":"If the two issues below are resolved, the paper makes a clean and substantial contribution: it gives an explicit, finite-codimensional description of the controllable subspace, independent of the time horizon and of the open control set, and it isolates the purely imaginary stationary modes as the exact obstruction. The real-Hilbert-space duality framework is natural and well executed, and the double-spectrum formalism for the measurable-product case is a promising approach. The open-set theorem is largely self-contained and, after correction of the identity in Eq. (3.9), appears sound. The measurable-product theorem, however, rests on an unproved multiplier approximation imported from preprints, so its status is conditional on that external input.","major_comments":[{"comment":"Equation (3.9) states Re(e^{it(Δ-V)}φ) = 1/2(e^{it(Δ-V)}φ + e^{-it(Δ-V)}φ). This is not an identity for complex-valued φ. Since H=-Δ+V has real coefficients, one has \\overline{e^{-itH}φ}=e^{itH}\\bar φ = e^{-it(Δ-V)}\\bar φ, so the second summand should contain \\bar φ, not φ. The same incorrect formula appears in Section 1.3. Lemma 3.6 uses (3.9) for φ=(I-Π_N)u_0, which is generally complex, so the proof of the high-frequency real-part estimate is not valid as written. The argument appears repairable: replacing φ by \\bar φ in the I_- term preserves the required bound because Π_N commutes with complex conjugation and the estimates depend only on the coefficient moduli. Nevertheless, this is a load-bearing error in the proof of Proposition 3.4 and Theorem 3.1, and it must be corrected.","section":"§1.3 and §3.1, Eq. (3.9)"},{"comment":"Proposition 4.4(iii) is the key multiplier approximation used in Lemma 4.5 and therefore in Theorem 4.3 and Theorem 1.2. The proof of Step 1 merely states that Burq-Zhu's Theorem 1.10, together with estimate (ii), puts χ=1_G in the multiplier-norm closure of trigonometric polynomials; it does not reproduce the exact statement or hypotheses of that theorem, nor the passage to (4.7)-(4.9). Since Lemma 4.5, Lemma 4.9, and Theorem 4.3 all inherit this approximation, Theorem 1.2 has no self-contained proof of its central input. The authors should either state and prove the needed closure result in detail or give a direct proof. If the cited preprints do not establish exactly (4.7), the measurable-product characterization is unsupported.","section":"§4.1, Proposition 4.4(iii)"}],"minor_comments":[{"comment":"The displayed spatial Fourier coefficient of v_{S^c} should be 1/2(a_k e^{-iµ_k t} + \\overline{a_{-k}} e^{iµ_k t}); the missing complex conjugation does not change the subsequent ℓ^2 estimate, but the formula as written is not the coefficient of Re(e^{it(Δ+c)}φ).","section":"§4.7, Step 3"},{"comment":"The replacement of ζ_0 by ζ=Re ζ_0 is justified by the pointwise bound |χ-Re ζ_0|=|Re(χ-ζ_0)|≤|χ-ζ_0| because χ is real-valued; this is true, but it should be stated explicitly since the displayed argument only says that the replacement does not increase the error.","section":"§4.1, proof of Proposition 4.4(iii), Step 1"},{"comment":"The paper would benefit from an explicit statement of which theorem in [14] is used for Proposition 4.4(i) and which theorem in [14] or [15] is used for Proposition 4.4(iii), as the current references are only listed by preprint number.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The open-set theorem (Theorem 1.1) is the more self-contained contribution and is likely correct after fixing the identity in Eq. (3.9). The measurable-product theorem (Theorem 1.2) is conditional on Proposition 4.4(iii), which is imported without proof from Burq-Zhu preprints; the editors may wish to verify with the authors that the cited theorem indeed implies the stated multiplier approximation. The incorrect identity in Section 1.3 and Eq. (3.9) must be corrected before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper answers a genuinely natural question: when controls enter only through their real part, which states are lost? The answer for open cylindrical control sets, X = K_Im^{⊥R}, is clean and convincing: the obstruction is exactly the purely imaginary stationary modes, independent of T and E. Section 3 is self-contained, and I found the compactness–uniqueness argument with the finite-dimensional space N_T well executed. The optimality remark about the correction term is also correct. That part alone is a solid contribution.\n\nThe measurable-product case in Section 4 is a different matter. The double-spectrum idea is real: the real part couples forward and backward evolutions, and the high-frequency control via multiplier approximation is plausible. But the key input, Proposition 4.4(iii), is not proved in the paper. The text simply says it follows from Burq–Zhu's Theorem 1.10 and the mixed-norm estimate, without reproducing the exact statement or showing how it yields the shifted forward and backward bounds in (4.7)–(4.9). Since Lemma 4.5, Lemma 4.9, and Theorem 4.3 all inherit this approximation, Theorem 1.2 is not self-contained. This is a real soft spot, not a nitpick. If Proposition 4.4(iii) is not exactly what the cited preprints provide, the measurable-product characterization is unsupported.\n\nThere is also a displayed identity error: in Section 1.3 and Eq. (3.9), Re(e^{it(∆-V)}φ) is written as (e^{itA}φ + e^{-itA}φ)/2. The correct second branch is e^{-itA}φ̄. The stress-test note says the expansion is correct; I disagree with that specific formula, but I also agree the consequence is not load-bearing. With φ replaced by φ̄ in the backward branch, the subsequent high-frequency estimates still go through because the Fourier coefficients simply conjugate. So this is a repairable slip.\n\nWho should read this: people working on controllability of Schrödinger equations on tori, especially the subfield of real-valued or constrained controls. The open-set theorem deserves to be in the literature even if the measurable-product theorem still needs pinning down. I would send it to a serious referee, with a clear request: fix the identity, and either prove Proposition 4.4(iii) or quote the precise Burq–Zhu result and verify the shifted branches. Not ready for acceptance as is.","headline":"A clean structural answer for real-part controllability on tori, proven self-containedly in the open-set case; the measurable-product case rests on an unproved external multiplier approximation and one wrong conjugate identity.","tokens_in":34047,"tokens_out":3731,"would_cite":true,"duration_ms":36745,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q41","93B05","93B07","93C20","35B60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Schrödinger equations on tori controlled only through the real part of a forcing term, the initial states that can be driven to zero are exactly those real-orthogonal to the purely imaginary stationary modes.","keywords":["Schrödinger equation on tori","real-part controls","controllable subspace","null controllability","observability inequality","stationary modes","measurable product control sets","double spectrum"],"falsifier":"Exhibit one initial datum $\\phi\\notin K_{\\mathrm{Im}}$ for which $\\operatorname{Re}(e^{it(\\Delta-V)}\\phi)$ vanishes on $(0,T)\\times E$; the paper's Lemma 3.7 says no such datum exists, so any example would falsify Theorem 1.1. For the shifted free measurable-product case, the parallel test is to find $\\phi\\notin (K_c)_{\\mathrm{Im}}$ with $\\operatorname{Re}(e^{it(\\Delta+c)}\\phi)=0$ on an admissible product set $G$, which would falsify Theorem 1.2.","tokens_in":33021,"feed_emoji":"🎛️","tokens_out":24964,"duration_ms":218642,"temperature":0.7,"pith_summary":"For a Schrödinger equation on the $d$-dimensional torus whose control enters only through its real part, full null controllability can fail for a structural reason: the purely imaginary stationary modes of the operator are invisible to the control. The paper proves that this obstruction is the only one. For cylindrical control regions $(0,T)\\times E$ with $E$ nonempty and open, and for admissible measurable product regions in the shifted free case, the controllable subspace is exactly the real orthogonal complement of the set of purely imaginary stationary modes, and it does not depend on $T$ or on the particular control region. The proof proceeds by real Hilbert space duality, reducing controllability to observability inequalities for the real part of the Schrödinger flow with a sharp stationary correction term. If the characterization is correct, then whenever the operator has no stationary kernel, or in the shifted free case no stationary Fourier mode, real-part controls already give full null controllability even from arbitrarily small open sets.","feed_headline":"Only imaginary stationary modes block real-part control","feed_subtitle":"It identifies the lost directions exactly and shows the control region's geometry adds no further obstruction.","key_machinery":"The load-bearing structure is the real Hilbert space structure on $L^2(\\mathbb T^d;\\mathbb C)$ given by $\\langle u,v\\rangle_{\\mathbb R}=\\operatorname{Re}\\int u\\,\\overline v\\,dx$. The real-part observation couples the forward and backward Schrödinger evolutions into the double spectrum $\\Sigma_c=\\{(-\\mu_n,n),(\\mu_n,n):n\\in\\mathbb Z^d\\}$ with $\\mu_n=|n|^2-c$. An admissible measurable product set is a Cartesian product of measurable factors over blocks of variables of size one or two, each factor of positive measure. The duality proposition converts controllability into lower bounds for the real-part observation with the stationary projection as correction term. For open sets the proof combines a high-frequency oscillatory estimate, a compactness–uniqueness argument, and elliptic unique continuation to identify the invisible space with $K_{\\mathrm{Im}}$; for admissible measurable product sets, it uses the approximation of the rough indicator by real trigonometric polynomials, product stability under small removals, and non-resonant translations to absorb finitely many low frequencies one at a time.","core_discovery":"Let $H=-\\Delta+V$ with $V\\in C(\\mathbb T^d;\\mathbb R)$, let $K=\\ker H$, and let $K_{\\mathrm{Im}}=\\{i\\operatorname{Im}\\phi:\\phi\\in K\\}$. The paper's central claim is that for $G=(0,T)\\times E$ with $E\\subset\\mathbb T^d$ nonempty and open, the set of initial data that can be driven to zero by a real-part control is exactly the real orthogonal complement $K_{\\mathrm{Im}}^{\\perp_\\mathbb R}$, computed with the inner product $\\langle u,v\\rangle_{\\mathbb R}=\\operatorname{Re}\\int_{\\mathbb T^d} u\\,\\overline v\\,dx$. The same conclusion holds for the shifted free operator $-\\Delta-c$ when the control set is an admissible measurable product set, with the obstruction generated by the Fourier modes satisfying $|n|^2=c$. The paper expresses this as observability inequalities with stationary correction terms, and proves the correction is optimal: projecting onto any proper real subspace of $K_{\\mathrm{Im}}$ would make some nonzero invisible mode uncontrollable.","pith_inferences":["A natural extension the paper does not pursue: the same 'purely imaginary stationary modes are the only obstruction' principle should hold for any self-adjoint operator with compact resolvent whose control map is a real-linear projection, which would make the result an abstract theorem about unitary groups rather than a torus-specific one.","For the free equation, the single lost direction is the purely imaginary constant, so adding one independent real-valued control channel that moves the imaginary mean would restore full controllability; this is a concrete, testable corollary of the paper's characterization.","The double-spectrum mechanism relies on the quadratic dispersion $\\mu_n=|n|^2-c$; transferring it to fractional Schrödinger equations with $\\mu_n=|n|^s-c$ on tori would predict a stationary obstruction at $|n|^s=c$, which the paper does not address.","Because Theorem 1.2 depends on the imported multiplier approximation for rough sets, a computational check of that approximation for simple product sets would show where the measurable-set result can be extended or where it needs a new idea."],"forward_implications":["If $V\\ge 0$ with positive spatial mean, then $K=\\{0\\}$, so the correction term vanishes and every initial datum is null-controllable with real-part controls.","For $V=0$, the only uncontrollable direction is the imaginary part of the spatial average: $X=\\{\\phi:\\operatorname{Im}\\int_{\\mathbb T^d}\\phi\\,dx=0\\}$.","For $V\\equiv -c$, full null controllability holds if and only if $c\\notin\\{|n|^2:n\\in\\mathbb Z^d\\}$, because this is exactly the condition for the stationary Fourier space $K_c$ to be trivial.","Within each geometric class, the controllable subspace is independent of the control time and of the particular open or admissible measurable control region.","The stationary correction term in the observability inequality cannot be shrunk: every proper real subspace of $K_{\\mathrm{Im}}$ leaves some nonzero mode uncontrollable."],"supporting_citations":[{"why":"Supplies the classical complex-valued observability inequality for Schrödinger evolutions on tori, which is the high-frequency input in the open-set real-part estimate.","marker":"[3]"},{"why":"Provides the compactness–uniqueness strategy that the open-set proof adapts to real-part observations.","marker":"[5]"},{"why":"Supplies the complex-valued observability results from admissible measurable product sets and their companion measurable-time inputs, including the multiplier approximation imported as Proposition 4.4(iii).","marker":"[14, 15]"},{"why":"Provides the uncertainty-principle finite-difference mechanism used to absorb a single spacetime frequency in the double-spectrum argument.","marker":"[26]"},{"why":"Provides the eigenvalue-counting bound used to make the high-frequency real-part estimate quantitative.","marker":"[27]"},{"why":"Provides the elliptic unique continuation theorem used to identify the invisible space with the purely imaginary stationary modes.","marker":"[41]"}],"fun_headline_variants":["Real-part control fails exactly on imaginary stationary modes","Exact obstruction to real-part control: imaginary stationary modes","Only imaginary stationary modes are the controllability obstruction","Real-part control: only imaginary stationary modes are lost"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported approximation result used for rough control sets: the characteristic function of the control region can be replaced, up to an arbitrarily small error when applied to Schrödinger solutions, by a real trigonometric polynomial, and this paper relies on that replacement for the measurable-product theorem without proving it here.","fun_headline_variants_meta":{"raw":{"variants":["Real-part control fails exactly on imaginary stationary modes","Exact obstruction to real-part control: imaginary stationary modes","Only imaginary stationary modes are the controllability obstruction","Real-part control: only imaginary stationary modes are lost"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001327,"raw_usage":{"total_tokens":5395,"prompt_tokens":936,"completion_tokens":4459,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":4397}},"tokens_in":552,"tokens_out":4459,"duration_ms":28150,"temperature":1.0,"reasoning_tokens":4397,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T19:07:59.665166+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit one initial datum $\\phi\\notin K_{\\mathrm{Im}}$ for which $\\operatorname{Re}(e^{it(\\Delta-V)}\\phi)$ vanishes on $(0,T)\\times E$; the paper's Lemma 3.7 says no such datum exists, so any example would falsify Theorem 1.1. For the shifted free measurable-product case, the parallel test is to find $\\phi\\notin (K_c)_{\\mathrm{Im}}$ with $\\operatorname{Re}(e^{it(\\Delta+c)}\\phi)=0$ on an admissible product set $G$, which would falsify Theorem 1.2.","supporting_citations":[{"cited_title":"Anantharaman and F","cited_arxiv_id":null,"evidence_quote":"Supplies the classical complex-valued observability inequality for Schrödinger evolutions on tori, which is the high-frequency input in the open-set real-part estimate."},{"cited_title":"Bardos, G","cited_arxiv_id":null,"evidence_quote":"Provides the compactness–uniqueness strategy that the open-set proof adapts to real-part observations."},{"cited_title":"Havin and B","cited_arxiv_id":null,"evidence_quote":"Provides the uncertainty-principle finite-difference mechanism used to absorb a single spacetime frequency in the double-spectrum argument."},{"cited_title":"Huang and C","cited_arxiv_id":null,"evidence_quote":"Provides the eigenvalue-counting bound used to make the high-frequency real-part estimate quantitative."},{"cited_title":"Schechter and B","cited_arxiv_id":null,"evidence_quote":"Provides the elliptic unique continuation theorem used to identify the invisible space with the purely imaginary stationary modes."}],"review_version":1}