{"id":"ce37dc2a-0662-4ac2-953c-4ef1978c524b","arxiv_id":"2411.15792","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new Carleman inequality yields Lipschitz (local) and logarithmic (global) stability estimates for an inverse obstacle problem for the magnetic Schrödinger equation.","lead":"The paper proves that the unknown boundary data of a magnetic Schrödinger equation on an obstacle can be stably recovered from measurements at a surrounding boundary, with Lipschitz stability on short time windows and logarithmic stability globally. It is a theoretical contribution to inverse problems, establishing a new Carleman inequality for the dynamic Schrödinger operator.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.1's Carleman estimate is proved only for a=0, and the absorption of magnetic-potential terms is asserted rather than shown; Theorem 1.1 depends on this unverified step.","rationale":"The reader's weakest-assumption analysis identifies exactly the step I consider most load-bearing: Proposition 2.1 is the engine of Theorem 1.1, and its proof is carried out only for the zero-magnetic-potential operator. The sentence 'Due to the large parameters γ and s...' is an assertion that a perturbation argument exists; it is not the argument. All subsequent estimates in Section 2—the L² bound for f on Γ×(ε,T−ε), the H¹ bound via (2.1), and the global logarithmic estimate—inherit this gap. I do not see an independent error in the a = 0 Carleman calculation, in the choice of the weight φ (provided m is chosen so that the weight is negative on D, an implicit but standard condition), or in the application of the estimate to the inverse problem. The additional boundary terms introduced by a are likely absorbable for large γ and s because σ grows with sγξ and the inner-boundary term has the correct sign, but this needs to be demonstrated explicitly. Because the gap is localized and plausibly fixable, the reader's CONDITIONAL verdict remains appropriate; no verdict change is needed.","tokens_in":10639,"tokens_out":16041,"duration_ms":148605,"concrete_test":"Write out the full identity for P_s^a z with nonzero a in Section 3, explicitly displaying every added bulk and boundary term relative to the a = 0 case. Then verify term-by-term that the added terms are bounded by ε times the left-hand side of Proposition 2.1 for γ ≥ Γ(||a||_{W^{3,∞}}) and s ≥ S(||a||_{W^{3,∞}}), with particular attention to the boundary integral on Σ: check that no uncontrolled term of the form s|a||∂ν z||z| or s|a||∂ν z|² survives after the signed boundary term σ(|∂ν z|² + σ²|z|²) is applied. Insert this as a lemma before Proposition 2.1; if every term absorbs, the theorem's central estimate stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 2.1 (Section 3) begins with the assertion: 'Due to the large parameters gamma and s, it is enough to establish the expected inequality when a = 0.' This is the only place where the magnetic potential is handled in the main Carleman estimate, and it is not demonstrated. The subsequent integration-by-parts computation is carried out for P = i∂t + Δ_g, not for the magnetic operator P = i∂t + L_a. Passing from a = 0 to general a requires controlling the additional terms e^{sϕ}(L_a − Δ_g)e^{−sϕ}z, which are of the form 2i a·∇z − 2is(a·∇ϕ)z + V z, plus derivatives of a. In the bilinear estimate, these produce bulk terms of size s|a||∇z|², s³|a||∇ϕ||∇z||z|, and s|V||z|², and, more delicately, boundary integrals on Σ involving ∂ν z with coefficients depending on a. Absorption into the left-hand side requires σ = sγξ to dominate |a| uniformly and requires the signed boundary term on Σ to control the new normal-derivative terms. Large γ and s plausibly suffice, but this is not a one-line consequence of the a = 0 estimate: one must prove the perturbation does not destroy the positivity structure on the inner boundary. Since Theorem 1.1's constants are C(ζ) with ζ containing a, if this absorption fails the main stability estimate is unsupported. The gap is real, localized, and fixable, but it should be written out before the proof is accepted as complete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes stability inequalities for an inverse obstacle problem for the magnetic Schrödinger equation on an exterior-type domain. The unknown quantity is the Dirichlet trace f on the obstacle boundary Γ×(0,T), and the data are the trace and normal derivative of the solution on an outer boundary Σ0. The main result, Theorem 1.1, gives a Lipschitz stability estimate on the time interval (ε,T−ε) with constant Ce^{c/ε}, and Corollary 1.3 converts this into a global logarithmic stability estimate via an external Hardy-type inequality. The proof relies on a Carleman estimate for the magnetic Schrödinger operator with a degenerate weight, Proposition 2.1, whose proof occupies Section 3, and on a well-posedness result for the forward IBVP in Appendix A.","tokens_in":11006,"tokens_out":10756,"duration_ms":103205,"significance":"If the proof is completed, this appears to be the first quantitative stability result for an inverse obstacle problem for the magnetic Schrödinger equation, and the Carleman estimate with boundary terms involving only time and tangential derivatives on the inner boundary is potentially reusable. The proof is largely self-contained, with the main external ingredient being a Hardy-type inequality from the authors' prior work [2,5], which is independently published. The paper also gives a clean semigroup argument for well-posedness. However, the central Carleman estimate is currently not fully justified for the magnetic operator, so the main theorem is not yet supported as written.","major_comments":[{"comment":"The Carleman estimate is proved only for a=0. The sentence 'Due to the large parameters γ and s, it is enough to establish the expected inequality when a = 0' is asserted but not demonstrated. For the magnetic operator P = i∂t + Δ_g + 2ia·∇ + V_a, conjugation with e^{sϕ} produces additional terms of the form 2ia·∇z − 2s(a·∇ϕ)z + V_a z, where V_a contains first derivatives of a and the quadratic term in a. In the bilinear estimates these give bulk terms of size s|a||∇z|², s³|a||∇ϕ||∇z||z|, and s|V_a||z|², and, more delicately, boundary integrals on Σ involving ∂ν z with coefficients depending on a. Absorbing these terms requires σ = sγξ to dominate |a| uniformly and requires the signed boundary term on Σ to control the new normal-derivative contributions. Because the constants in Theorem 1.1 depend on a through ζ, this absorption must hold uniformly in the admissible class; otherwise the main stability estimate is unsupported. This gap is localized and fixable, but the perturbation calculation must be written out before the proof can be accepted.","section":"Section 3, proof of Proposition 2.1"},{"comment":"The parameter m appearing in ϕ(x,t) = (e^{γ(φ(x)+2m)} − e^{4γm})ℓ(t) and ξ(x,t) = e^{γ(φ(x)+2m)}ℓ(t) is never defined. In the proof of Theorem 1.1, the estimate e^{2sϕ}ω³ ≤ C e^{−csℓ} on Σ0 is essential for bounding the boundary-data terms by the data norms. This estimate is true only if m is chosen so that φ < 2m on ∂Ω, making ϕ negative on Σ0. Without an explicit definition or condition on m, the stated upper bound does not follow and the final stability inequality is not justified. Please add the required condition on m and include m in the parameter set ζ if needed.","section":"Section 2, definition of the weight functions"},{"comment":"In the derivation of I3, the identity I3 = (1/2)∫_Q sϕ′′|z|² is obtained by integrating by parts in t and dropping the boundary terms at t=0 and t=T. Since ϕ ∼ ℓ(t) and ℓ is singular at the endpoints, these boundary terms vanish only if z decays sufficiently fast as t→0,T, for instance because e^{sϕ}→0 at the endpoints. This requires the same condition on m as in the previous comment and should be stated explicitly. For the function class u ∈ L²((0,T);H²(D)) ∩ H¹((0,T);H¹(D)) with no vanishing condition at t=0,T, the integration by parts is not automatically justified.","section":"Section 3, integration by parts in time"}],"minor_comments":[{"comment":"There is a typo in 'non-homogenuous' on page 2; it should be 'non-homogeneous'.","section":"Introduction"},{"comment":"The Hardy-type inequality from [5, Corollary 3.1] is cited rather than stated. Since the corollary is load-bearing for the global logarithmic stability, it would improve the paper to state the precise inequality and the hypotheses needed for its application to f.","section":"Corollary 1.3"},{"comment":"When passing from the z-variable to u = e^{−sϕ}z at the end of the proof of Proposition 2.1, the cross terms generated by z′ = e^{sϕ}(u′ + sϕ′u) and ∂ν z = e^{sϕ}(∂ν u + s∂νϕ u) are not displayed. These are standard to absorb using the large parameters, but a short justification would make the proof fully detailed.","section":"Section 3, final substitution"},{"comment":"The uniqueness claim for the IBVP with f=0 and u0=0 is asserted after the semigroup construction; it would be clearer to state explicitly that it follows from the contraction semigroup property, although this is standard.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The main gap is localized to the proof of Proposition 2.1, specifically the reduction to a=0 and the handling of the magnetic potential. This is fixable by a direct perturbation computation, and the rest of the paper is in good shape. The self-citations [2,5] are to independently published results and do not raise circularity concerns. I would support acceptance after the magnetic perturbation argument and the condition on m are supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine first — stability for an inverse obstacle problem for the magnetic Schrödinger equation — and most of the machinery is assembled with care. But the key Carleman estimate contains one real gap: the magnetic potential is dismissed by fiat (“due to the large parameters γ and s...”), and the computation that follows is for a = 0. The reduction is probably fixable, but it is asserted, not demonstrated, and Theorem 1.1 sits on it.\n\nWhat is actually new: the boundary-to-boundary stability estimates for P = i∂_t + L_a, with exponential-in-1/ε Lipschitz-type control locally in time and a logarithmic modulus globally via the interpolation argument. The Carleman inequality in Proposition 2.1, featuring time-derivative and tangential-derivative terms on the inner boundary, is new for the dynamic Schrödinger operator, and its derivation from the nine I_k terms is mostly clean algebra. The step from the Carleman estimate to Theorem 1.1 is the right argument: negative weight on the obstacle boundary, admissible-class control of ∂_t f and ∇_τ f by (β/α)‖f‖, and the ℓ(t) ≥ Tε/2 cut. The semigroup well-posedness appendix is standard and fine.\n\nWhere I push back: the a = 0 reduction is the one spot I want written out. The magnetic perturbation contributes bulk terms of size s‖a‖|∇z|² and s³‖a‖|∇φ||∇z||z|, plus lower-order terms, and — more delicately — boundary integrals on Σ with ∂_ν z weighted by a. Large γ and s plausibly absorb all of this, since σ = sγξ can dominate ‖a‖ uniformly, but the positivity structure on the inner boundary is the delicate part of the estimate, and that is exactly where the paper says “enough” instead of showing it. Because the constants in Theorem 1.1 carry the W^{3,∞} norm of a, the absorption has to be uniform. I would not call this fatal; I would call it a necessary expansion.\n\nMinor: Corollary 1.3 imports a Hardy-type inequality from the authors’ own prior works [5] and [2]. Those are independently published results, not the target theorem, so the self-citation is not a problem; just note the global result leans on it.\n\nAudience: people doing Carleman-based stability for Schrödinger equations or inverse obstacle problems. It deserves a serious referee, with the request to expand the magnetic absorption. I agree with the conditional verdict, leaning positive.","headline":"A solid, genuinely new stability result for the magnetic Schrödinger inverse obstacle problem; the core Carleman estimate has one asserted-but-unshown step (the a=0 reduction) that should be expanded before acceptance.","tokens_in":11468,"tokens_out":9744,"would_cite":true,"duration_ms":80405,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35Q41","58J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two boundary measurements determine magnetic Schrödinger obstacle data","keywords":["inverse obstacle problem","magnetic Schrödinger equation","Lipschitz stability","logarithmic stability","Carleman inequality","boundary measurements","Carleman estimate with degenerate weights","Cauchy data"],"falsifier":"Perform the missing absorption step for a nonzero magnetic potential: take a constant $a$ in Euclidean space with a simple annulus geometry and track every term involving $a$ through the splitting of the conjugated operator. If any magnetic term survives at order comparable to the leading Carleman weight, for instance a term of order $s^2\\gamma^2\\xi^2|z|^2$ or a boundary term not controlled by $\\sigma(|\\partial_{\\nu}z|^2+\\sigma^2|z|^2)$, then the stated inequality with constants independent of $a$ fails. A direct numerical evaluation of Proposition 2.1 with constant $a$ and a smooth exact solution would be a concrete test.","tokens_in":10448,"feed_emoji":"🧲","tokens_out":8738,"duration_ms":73433,"temperature":0.7,"pith_summary":"This paper establishes quantitative stability for an inverse obstacle problem for the magnetic Schrödinger equation: the unknown function $f$ prescribed on the boundary $\\Gamma$ of an obstacle is determined, with explicit error bounds, from two measurements on a surrounding surface $\\partial\\Omega$, namely the solution $u(f)$ and its normal derivative $\\partial_{\\nu_g}u(f)$. The main theorem is a Lipschitz-type estimate locally in time, $$\\|f\\|_{$H^{1}$(\\Gamma\\times(\\varepsilon,T-\\varepsilon))}\\le C $e^{{c/\\varepsilon}}$\\left(\\|u(f)\\|_{$H^{1}$(\\Sigma_0)}+\\|\\partial_{\\nu_g}u(f)\\|_{$L^{2}$(\\Sigma_0)}\\right),$$ valid for every admissible $f$ and every $\\varepsilon\\in(0,T/2)$. A global-in-time $L^2$ estimate follows by interpolation and gives a logarithmic modulus of continuity. This matters because it turns uniqueness into stability: small changes in the far-field measurements force small changes in the obstacle boundary data, with the control deteriorating exponentially as the observation window approaches the initial and final times. The authors state this is the first stability result for this inverse obstacle problem for the magnetic Schrödinger equation.","feed_headline":"Two boundary measurements determine magnetic Schrödinger obstacle data","feed_subtitle":"Stable recovery of unknown obstacle data from two Schrödinger boundary measurements.","key_machinery":"The carrying object is a Carleman estimate for the dynamical magnetic Schrödinger operator $P=i\\partial_t+L$ on the space-time cylinder $D\\times(0,T)$, with the degenerate weight $\\phi(x,t)=(e^{\\gamma(\\varphi(x)+2m)}-e^{4\\gamma m})\\ell(t)$ and $\\ell(t)=[t(T-t)]^{-1}$; here $\\varphi$ is a $C^4$ function whose positive level set has no critical points and satisfies a strong convexity condition, so the weight blows up at the initial and final times. The estimate controls weighted first-order norms of $u$ in the interior and weighted normal-derivative data on the inner boundary $\\Sigma$ in terms of $\\|Pu\\|^2$ plus boundary terms on the outer boundary $\\Sigma_0$ and tangential/time-derivative terms on $\\Sigma$. In the proof the conjugated operator is split into self-adjoint and skew-adjoint parts and the cross term is expanded; the large parameters $\\gamma$ and $s$ absorb lower-order terms, and the magnetic potential $a$ enters as a lower-order perturbation. The admissible-set bounds on $f$ are then used to absorb the time and tangential derivatives of $u$ on $\\Gamma$ into the diagonal Carleman weight on the left-hand side. The same machinery, together with a Hardy-type inequality, yields the global logarithmic version.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.1 and the Carleman mechanism behind it. For the admissible class $F$ of boundary functions $f$ satisfying a lower bound on their $L^2$ norm and an upper bound on time and tangential derivatives, the two boundary measurements $(u(f)|_{\\Sigma_0},\\partial_{\\nu_g}u(f)|_{\\Sigma_0})$ dominate $f$ on the inner boundary $\\Gamma$ over any open time interval strictly inside $(0,T)$. The inequality has the explicit form displayed above; it is local in time because intervals near $t=0$ and $t=T$ are discarded. Two corollaries are drawn directly: if $f(x,t)=a(x)b(t)$ with $b$ known, then $a$ is Lipschitz stable on $\\Gamma$ from the same measurements; and by a Hardy-type inequality the full time interval is recovered with a logarithmic modulus of continuity. The proof route is a Carleman estimate for $i\\partial_t+L$, followed by an absorbing argument on the inner boundary that uses the admissible-class bounds to discard lower-order terms.","pith_inferences":["Because the magnetic terms are treated as lower-order perturbations, the same proof route should extend to time-dependent magnetic potentials or to lower-order complex potentials, provided the same absorption can be made uniform.","The explicit $e^{c/\\varepsilon}$ factor signals severe ill-posedness near $t=0$ and $t=T$; a natural next step is to check whether this rate is optimal by constructing explicit solutions, for example with a flat metric and a constant magnetic field, whose boundary traces saturate the estimate.","The Carleman inequality itself may be reusable as a black box for other inverse problems for the magnetic Schrödinger equation, such as recovering a magnetic or electric potential from boundary observations, since the weight is independent of $a$."],"forward_implications":["For every $\\varepsilon\\in(0,T/2)$, the Cauchy data on the outer boundary determine $f$ on $\\Gamma\\times(\\varepsilon,T-\\varepsilon)$, with the error growing at most like $e^{c/\\varepsilon}$ as $\\varepsilon\\to 0$.","When $f$ separates as $a(x)b(t)$ with $b$ known, the spatial factor $a$ is recovered Lipschitz-continuously on the whole obstacle boundary $\\Gamma$.","Globally in time, $f$ is determined from the same two measurements with a logarithmic modulus of continuity, not merely a uniqueness statement.","The two-measurement inverse obstacle strategy, previously applied to elliptic, parabolic, and hyperbolic equations, now covers the magnetic Schrödinger equation as well."],"supporting_citations":[{"why":"Supplies the admissible weight function $\\varphi$ used in the Carleman estimate and the preceding hyperbolic inverse obstacle problem that this work extends.","marker":"[3]"},{"why":"Establishes the two-measurement inverse obstacle strategy for elliptic and parabolic equations, the direct predecessor of the present stability estimate.","marker":"[4]"},{"why":"Provides the Hardy-type inequality used to control the time intervals near $t=0$ and $t=T$ in the global logarithmic stability corollary.","marker":"[5]"},{"why":"Supplies the companion Hardy-type inequality cited together with [5] for the endpoint interpolation estimate.","marker":"[2]"},{"why":"Provides the semigroup well-posedness facts used in Appendix A to construct the solution $u(f)$ for admissible boundary data $f$.","marker":"[9]"}],"fun_headline_variants":["Two boundary measurements stabilize magnetic Schrödinger obstacle recovery","Stable recovery of obstacle data from two Schrödinger boundary measurements","Two boundary measurements give Schrödinger obstacle stability","Schrödinger obstacle recovery stable from two boundary observations","Lipschitz stability for Schrödinger obstacle from two boundary data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central estimate is proved only for the magnetic potential $a=0$; the paper asserts that the large parameters $\\gamma$ and $s$ make the magnetic terms harmless, but it does not display the absorption calculation, and the final constants depend on $a$ and must remain uniform for the theorem to hold.","fun_headline_variants_meta":{"raw":{"variants":["Two boundary measurements stabilize magnetic Schrödinger obstacle recovery","Stable recovery of obstacle data from two Schrödinger boundary measurements","Two boundary measurements give Schrödinger obstacle stability","Schrödinger obstacle recovery stable from two boundary observations","Lipschitz stability for Schrödinger obstacle from two boundary data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000584,"raw_usage":{"total_tokens":2679,"prompt_tokens":808,"completion_tokens":1871,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":1793}},"tokens_in":424,"tokens_out":1871,"duration_ms":13414,"temperature":1.0,"reasoning_tokens":1793,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:53:04.463486+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform the missing absorption step for a nonzero magnetic potential: take a constant $a$ in Euclidean space with a simple annulus geometry and track every term involving $a$ through the splitting of the conjugated operator. If any magnetic term survives at order comparable to the leading Carleman weight, for instance a term of order $s^2\\gamma^2\\xi^2|z|^2$ or a boundary term not controlled by $\\sigma(|\\partial_{\\nu}z|^2+\\sigma^2|z|^2)$, then the stated inequality with constants independent of $a$ fails. A direct numerical evaluation of Proposition 2.1 with constant $a$ and a smooth exact solution would be a concrete test.","supporting_citations":[{"cited_title":"An inverse hyperbolic obstacle problem","cited_arxiv_id":"2407.05662","evidence_quote":"Supplies the admissible weight function $\\varphi$ used in the Carleman estimate and the preceding hyperbolic inverse obstacle problem that this work extends."},{"cited_title":"Choulli and M","cited_arxiv_id":null,"evidence_quote":"Provides the Hardy-type inequality used to control the time intervals near $t=0$ and $t=T$ in the global logarithmic stability corollary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the companion Hardy-type inequality cited together with [5] for the endpoint interpolation estimate."},{"cited_title":"Tucsnak and G","cited_arxiv_id":null,"evidence_quote":"Provides the semigroup well-posedness facts used in Appendix A to construct the solution $u(f)$ for admissible boundary data $f$."}],"review_version":1}