{"id":"d957e5cf-831c-4381-ab86-073ef82fae3d","arxiv_id":"2505.24368","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"When a potential grows at infinity, uniqueness for the parabolic Schrödinger equation holds under a weaker integral condition on the solution, with the improvement set by the decay rate of positive stationary solutions.","lead":"This paper proves new uniqueness conditions for solutions of the parabolic Schrödinger equation on complete noncompact Riemannian manifolds, showing that a growing potential allows uniqueness under a much weaker integrability requirement on the solution. The relaxation is achieved by weighting the solution with the decay of a positive solution to the associated stationary Schrödinger equation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharpest relaxation result Theorem 3.6 rests on an unproved manifold transfer of Agmon's comparison bound [4, Cor. 2.8]; if that estimate does not extend, the L2 ground-state decay underlying (3.9) is unsupported.","rationale":"The reader's conditional verdict is sound. The main internal flaw I checked—the algebra in Proposition 4.2—is repairable: using test function η=φψ in the subsolution inequality and η=φwψ in the supersolution inequality gives exactly the required B≥0, so it is not the load-bearing issue. The decisive point is the Agmon comparison on manifolds. The paper needs it precisely at the transition from the explicit radial supersolution z to the actual positive L2 solution φ, and no proof is supplied beyond an assertion of 'same proof'. A referee could not verify the central decay estimate without seeing that proof. The proposed H^n radial calculation is a minimal nontrivial case in which m>0 and the geometry is explicit; it would discriminate between a harmless transfer and a false one. Since the concern is a missing proof rather than a demonstrated contradiction, conditional rather than reject is right.","tokens_in":16376,"tokens_out":19474,"duration_ms":257829,"concrete_test":"On H^n (n≥2), take V(r)=c0 r^α + c1 for large r and smooth nonnegative inside, choose c1 so that inf spec(-Δ+V)=0, and compute the radial ground state φ by solving φ''+(n-1)coth(r)φ'-(c0 r^α+c1)φ=0 with φ(0)=1, φ'(0)=0. Check whether φ(r)exp(a0 r^(α/2+1)) is bounded as r→∞; if unbounded for some α>2, Proposition 6.5 is false. If bounded, repeat on a non-radial model manifold with m(r,θ)≥0, and/or rederive [4, Cor. 2.8] in polar coordinates to confirm that no Euclidean Fourier step is essential.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6.5 is the only place where the exponential rate a0 in (3.9) is justified. It asserts that, under (3.6), (3.7) and Λ1=0, the positive L2 solution φ of (1.7) satisfies φ≤C exp(-a0r^(α/2+1)). The proof uses compact resolvent to obtain φ∈L2, then invokes [4, Corollary 2.8] through Remark 6.4, which states that a result proved in R^n 'with the same proof also holds in M'. No hypotheses of [4, Cor. 2.8] are stated, and no manifold version is proved. The condition m(r,θ)≥0 guarantees that z is a supersolution in Proposition 6.3, but it does not by itself show that the actual ground state is bounded by that supersolution; Agmon's comparison in R^n uses Euclidean structure in ways that are not demonstrated to be dispensable. If the comparison fails on some complete manifold with a pole and m≥0, then Theorem 3.6 is not established: (3.9) would not imply (1.8), so the main relaxation claim does not follow. The alternative Theorem 3.8 only covers nonnegative subsolutions via a supersolution and does not recover Theorem 3.6. The reliance on [20] for the weighted heat uniqueness theorem is a second open thread, but it is at least stated rather than asserted to hold by 'same proof'.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies uniqueness for the Cauchy problem associated with the parabolic Schrödinger equation on complete noncompact Riemannian manifolds, under integral conditions on the solution. The main mechanism is a change of variable w = u/φ, where φ is a positive solution of the stationary Schrödinger equation Δφ - Vφ = 0; this transforms solutions of the parabolic problem into solutions of a weighted heat equation, reducing uniqueness to known or cited weighted-heat uniqueness theorems. The paper's central claim is that the standard uniqueness condition ∫∫|u|^p e^{-h} dν < ∞ can be relaxed to ∫∫|u|^p e^{-h} φ^{2-p} dν < ∞, and that for potentials V(x) ≥ c0 r^α on manifolds with a pole and m(r,θ) ≥ 0, the decay of φ can be quantified as φ ≤ C exp{-a0 r^{α/2+1}}, yielding the concrete relaxed condition (3.9). The paper also establishes a version for nonnegative subsolutions using only weak supersolutions of the stationary equation, and it gives several explicit examples, including a sign-changing potential on R^n and model manifolds.","tokens_in":16645,"tokens_out":6370,"duration_ms":78253,"significance":"If the main theorem is correct, the paper makes a genuine contribution: it shows that the potential can enlarge the uniqueness class for the parabolic Schrödinger equation, with an explicit rate a0 = 2√c0/(α+2) entering the relaxed integrability condition. The reduction of the parabolic problem to a weighted heat equation via a positive solution of the stationary equation is clean, and the explicit computations in Sections 6 and 7 give verifiable estimates with no fitted parameters. The paper is also honest about the limitations of the supersolution method in Remark 6.8. However, the sharpest result for general solutions, Theorem 3.6, depends on an unproved manifold transfer of an Agmon comparison bound, and the main weighted-heat uniqueness theorem for 1 < p < 2 is deferred to a paper 'in preparation'. These dependencies make the central claim not yet fully verifiable in the present manuscript.","major_comments":[{"comment":"The proof of Theorem 3.6 rests entirely on the assertion in Remark 6.4 that the Agmon comparison bound [4, Corollary 2.8], proved in R^n, 'with the same proof also holds in M'. This transfer is load-bearing: Proposition 6.5 uses it to conclude that the L2 ground state φ of (1.7) satisfies φ ≤ C exp{-a0 r^{α/2+1}} on M∖B_R, which is what converts the supersolution of Proposition 6.3 into the relaxed condition (3.9). No hypotheses from [4, Corollary 2.8] are stated, and no manifold version is proved. Because Agmon's comparison argument may depend on Euclidean structure, the claim 'with the same proof' is not verifiable as written. Please either prove the manifold version under explicit geometric hypotheses, or state precisely which conditions on (M,g) and V are needed, or restrict Theorem 3.6 to the class where the comparison can actually be established.","section":"§6.1, Remark 6.4 and Proposition 6.5"},{"comment":"The main engine for all results is the weighted-heat uniqueness theorem, but for 1 < p < 2 it is attributed to [20], which is 'in preparation', and Theorem 5.2 is said to follow by 'minor changes' without proof. Since Theorem 3.1 and Theorem 3.3 reduce directly to these statements, the paper's central uniqueness claims are not self-contained. The authors should either state and prove the needed weighted-heat uniqueness results, or provide a precise statement of the result in [20] with all hypotheses and a reference that is publicly available, so that the reduction in Section 5 is verifiable.","section":"§5, Theorem 5.1"}],"minor_comments":[{"comment":"The equation numbering is inconsistent: the Green identity is numbered (2.1) after an equation already numbered (2.2), and the weighted Laplacian is also numbered (2.2). Please renumber.","section":"§2, equations (2.1)–(2.2)"},{"comment":"In Theorem 5.1 the solution is called w, but condition (5.1) and the conclusion use u; the same issue appears in Theorem 5.2. Please correct the notation so that the integrability condition and the conclusion refer to the same function.","section":"§5, Theorems 5.1 and 5.2"},{"comment":"In Example 7.1(b), the supersolution is defined as u(x,t) = exp{A(1-Qt)|x|^b}, but a few lines later θ(x,t) is set to A(1+Qt)r^b, and the computation of θ_t uses the sign AQ r^b instead of -AQ r^b. This inconsistency should be fixed; the supersolution construction may still work with the correct sign, but as written the computation is not correct.","section":"Example 7.1(b)"},{"comment":"In the discussion of the case γ = b, the line 'V(r) → sgn(ab − (n−1)αγ)∞' is ambiguous; please state explicitly the sign of the leading coefficient and the resulting limit.","section":"Example 7.2"},{"comment":"The extension of φ(x)=|x|^β_- from |x|>1 to all of R^n is described only as 'using a cutoff or mollification'; since β_- is negative, the extension near the origin must be made carefully to keep φ positive and C^2, and the resulting V must still be C^1. Please provide sufficient detail.","section":"Example 7.5"},{"comment":"There is a typo in 'corollariy' in Corollary 3.2. In Remark 3.7, the compact-resolvent implication is cited to [28, Theorem XIII.16], which is stated for R^n; a sentence justifying the manifold version would be helpful, as the same 'exactly the same holds on M' transfer appears here as in Remark 6.4.","section":"Corollary 3.2 and Remark 3.7"}],"recommendation":"major_revision","confidential_remarks":"The main theorem's dependence on an 'in preparation' paper [20] and on an unproved manifold extension of [4, Corollary 2.8] are the two concerns that most affect the manuscript's verifiability. I would recommend asking the authors to provide a proof or a precise public reference for the weighted-heat uniqueness theorem and to either prove or clearly delimit the Agmon comparison transfer before the paper is accepted. The examples in Section 7 are useful but contain a sign inconsistency that needs correction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'll get straight to it. The core idea is new: replace the usual weighted integrability condition in the parabolic Schrödinger uniqueness theorem with one that also includes the decay of a positive stationary solution φ, i.e. use φ^{2-p} as an extra weight. For potentials growing like r^α, with φ ~ exp(-a r^{α/2+1}), the condition genuinely weakens when α>2. The gauge transform u=φw is standard, so Theorem 3.1 is a corollary of known weighted heat uniqueness. The real work is in Section 6 and the examples: explicit supersolutions, decay estimates, and cases where the relaxation does and does not occur. Example 7.1 is particularly nice: it gives an explicit sign-changing V with φ=e^{-a r^b}, proves uniqueness under the relaxed condition, and even shows existence of a solution with exponentially growing initial data.\n\nSoft spots: Proposition 4.2's written proof has a coefficient error. The weak supersolution inequality is multiplied by φ, but (4.5) omits the φ^{-1} factors that should appear in the dµ conversion. The fix is straightforward, but as written the proof is not correct. More importantly, Theorem 3.6's sharp relaxation rests on Proposition 6.5, which invokes Agmon's comparison bound [4, Corollary 2.8] on R^n and asserts, in Remark 6.4, that it 'with the same proof' holds on a manifold with a pole and m≥0. No proof or precise hypotheses are provided. This is a load-bearing step: without it, you only have the supersolution z, which gives Theorem 3.8 for nonnegative subsolutions, not the general solution case of Theorem 3.6. The third soft spot is the central weighted heat uniqueness theorem, credited to [20], which is in preparation.\n\nOverall, the framework is coherent and the examples are instructive. These are gaps that can be fixed, not signs the idea is wrong. But the current manuscript is not fully verified.\n\nRecommendation: send it to a serious referee. The referees should be asked to prove the manifold version of the Agmon comparison (or give explicit counterexamples), correct the Prop 4.2 calculation, and either complete or replace the [20] reliance. If those come through, the paper is a real contribution.","headline":"The relaxed-uniqueness idea is clean and the explicit examples are solid, but the sharpest theorem depends on an unproved manifold version of Agmon's comparison and on an 'in preparation' reference.","tokens_in":17209,"tokens_out":6107,"would_cite":true,"duration_ms":67718,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K15","35J10","35A02","35A01","58J05","58J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A growing potential in the parabolic Schrödinger equation relaxes the integral condition that forces the zero solution, by replacing the usual heat-equation weight with the decay of a positive stationary solution.","keywords":["Schrödinger equation","parabolic Cauchy problem","uniqueness","Riemannian manifolds","weighted heat equation","sub–supersolutions","manifolds with a pole","exponential decay"],"falsifier":"On a model manifold with radial Laplacian coefficient $m(r) \\ge 0$ and potential $V = c_0 r^\\alpha$, compute the decaying radial solution $\\varphi$ of $\\varphi'' + m(r)\\varphi' - c_0 r^\\alpha \\varphi = 0$ and test whether $\\varphi(r)/\\exp\\{-a_0 r^{\\alpha/2+1}\\}$ stays bounded; if it is unbounded for some admissible $m$, Proposition 6.5 and the relaxed uniqueness condition fail.","tokens_in":16116,"feed_emoji":"⚛️","tokens_out":13680,"duration_ms":142292,"temperature":0.7,"pith_summary":"The paper establishes that, for the parabolic Schrödinger equation $u_t - \\Delta u + V u = 0$ on a complete noncompact Riemannian manifold, the presence of a suitable potential $V$ genuinely widens the uniqueness class for the zero-initial-value Cauchy problem. Where the heat-equation theory asks for $\\int_0^T\\int_M |u|^p e^{-h(r)}\\,d\\nu\\,dt < \\infty$, the paper shows it suffices to have $\\int_0^T\\int_M |u|^p e^{-h(r)} \\varphi^{2-p}\\,d\\nu\\,dt < \\infty$, with $\\varphi$ a positive solution of the stationary equation $\\Delta\\varphi - V\\varphi = 0$. For potentials growing like $c_0 r^\\alpha$, the constructed decay $\\varphi \\sim \\exp\\{-a_0 r^{\\alpha/2+1}\\}$ makes the new condition strictly weaker than the potential-free one, so solutions that would escape the heat-equation condition are still forced to vanish. The proof works by the gauge change $w = u/\\varphi$, which turns the Schrödinger equation into a weighted heat equation for the measure $\\varphi^2\\,d\\nu$.","feed_headline":"Growing potentials relax uniqueness for Schrödinger Cauchy problems","feed_subtitle":"If the potential grows like r to the α, uniqueness survives even when the classical heat-equation condition fails.","key_machinery":"The load-bearing identity is the gauge transform $w = u/\\varphi$: whenever $\\varphi$ is a positive classical solution of $\\Delta\\varphi - V\\varphi = 0$ and $d\\mu = \\varphi^2\\,d\\nu$, a solution of the parabolic Schrödinger equation becomes a solution of the weighted heat equation $w_t - \\Delta_\\mu w = 0$ with $\\Delta_\\mu w = \\Delta w + 2\\langle \\nabla\\varphi/\\varphi, \\nabla w\\rangle$. If $\\varphi$ is only a positive weak supersolution, nonnegative subsolutions become weak subsolutions of the same weighted problem. The second ingredient is the explicit radial supersolution $z(r) = \\exp\\{-a_0 r^{\\alpha/2+1}\\}$ for $V \\ge c_0 r^\\alpha$; the choice $a_0 = 2\\sqrt{c_0}/(\\alpha+2)$ makes the leading term of $\\Delta z - Vz$ nonpositive, and a comparison bound on exponential decay, quoted from the literature, transfers this decay to the actual solution.","core_discovery":"The paper claims that uniqueness for problem (1.1) can be read off from the stationary equation rather than from the geometry alone. If $\\Lambda_1 = \\inf \\sigma(-\\Delta + V) \\ge 0$ and $\\varphi > 0$ solves $\\Delta\\varphi - V\\varphi = 0$, then any solution $u$ satisfying (1.8) is identically zero; if only a positive weak supersolution $\\xi$ is available, the same conclusion holds for nonnegative subsolutions under (3.4). For potentials with $V \\ge c_0 r^\\alpha$ outside a ball and with $\\Lambda_1 = 0$, Proposition 6.5 produces a positive $L^2$ solution $\\varphi$ obeying $\\varphi \\le C\\exp\\{-a_0 r^{\\alpha/2+1}\\}$, so condition (3.9) follows; the same rate is obtained when $V \\ge -\\lambda_1$ via a locally Lipschitz weak supersolution, yielding uniqueness of nonnegative subsolutions. The paper presents examples on $\\mathbb{R}^n$, on model manifolds, and on hyperbolic space where the relaxed condition is verified.","pith_inferences":["Inference: the natural general weight suggested by the proof is $\\exp\\{-a(2-p)\\rho(x)\\}$ with $\\rho$ the Agmon distance of $-\\Delta + V$; the paper only realizes the radial rate $r^{\\alpha/2+1}$.","Inference: the borderline case $\\alpha = 2$ is not treated; the method suggests only polynomial or logarithmic corrections there, and it would be informative to test whether uniqueness still holds with a weight $e^{-h}\\,r^{-\\beta}$.","Inference: making the Euclidean-to-manifold transfer of the comparison bound explicit under curvature or volume-growth hypotheses would extend Theorem 3.6 beyond the class of manifolds with a pole and $m \\ge 0$."],"forward_implications":["Every solution satisfying the relaxed condition (1.8), or the explicit condition (3.9), is identically zero, so at most one solution of the inhomogeneous Cauchy problem (3.3) can meet the same condition.","For potentials with $V \\ge c_0 r^\\alpha$ and $\\Lambda_1 = 0$, uniqueness is obtained under a weight with the extra factor $\\exp\\{-a_0(2-p)r^{\\alpha/2+1}\\}$; when $\\alpha > 2$ this is a strictly weaker hypothesis than the classical heat-equation condition (1.4).","The same gauge transformation yields uniqueness of nonnegative subsolutions when only a locally Lipschitz weak supersolution of the stationary equation is available, as in Theorem 3.8.","In the explicit examples, initial data growing like $\\exp\\{K|x|^b\\}$ with $b > 2$ and $K$ below the stationary decay rate still admit a unique solution, so the relaxed condition is genuinely usable."],"supporting_citations":[{"why":"Supplies the comparison bound on exponential decay of positive Schrödinger solutions that the paper extends to manifolds with a pole to conclude the constructed supersolution rate is also a bound on the true solution.","marker":"[4]"},{"why":"Gives the weighted-heat uniqueness theorem for $1