{"id":"a3f8c037-c392-406a-8246-a833a6baf443","arxiv_id":"2606.18212","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Uniqueness of time-dependent magnetic and electric potentials from the Dirichlet-to-Neumann map for a nonlinear dynamical Schrödinger operator, with full and partial data results under analyticity assumptions.","lead":"The paper proves that the Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials for a nonlinear dynamical Schrödinger operator under analyticity assumptions, for both full and partial boundary data. A smart generalist might read it to see how boundary measurements recover internal time-varying potentials in a quantum system.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption already isolates the two load-bearing technical hypotheses. With the full text now notionally available, nothing in the stated claim introduces a new or more fragile assumption that would alter the UNVERDICTED status.","tokens_in":1573,"tokens_out":243,"duration_ms":19507,"concrete_test":"Re-derive the well-posedness statement for the forward problem (existence, uniqueness, and Sobolev regularity) directly from the nonlinear magnetic Schrödinger equation without invoking the inverse-problem sections; confirm the claimed regularity index matches the expected loss from the magnetic term and nonlinearity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is uniqueness of time-dependent magnetic and electric potentials from the DN map (full and partial data) under analyticity assumptions, together with well-posedness of the forward nonlinear dynamical Schrödinger problem at optimal Sobolev regularity. The analyticity is used for continuation, a standard device in this literature; the forward regularity claim is the other technical pillar. No internal inconsistency, hidden circularity, or unsupported step is visible from the abstract and claim description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript studies inverse problems for a nonlinear dynamical Schrödinger operator with magnetic and electric potentials. Under suitable analyticity assumptions on the potentials, it claims to prove that the Dirichlet-to-Neumann map uniquely determines the time-dependent magnetic and electric potentials, both from full boundary data and from partial data (with potentials known near the boundary and Neumann measurements on arbitrarily small boundary subsets). It also claims to establish well-posedness of the forward nonlinear problem, obtaining optimal Sobolev regularity for the solutions.","tokens_in":1644,"tokens_out":381,"duration_ms":18035,"significance":"If the uniqueness theorems hold, the work would extend existing results on inverse problems for Schrödinger operators to the nonlinear dynamical setting with magnetic potentials and would strengthen partial-data results via analytic continuation. The forward well-posedness statement with optimal regularity would provide a useful technical foundation for such inverse problems. The analyticity assumption is a standard device in this literature and is used explicitly for continuation arguments.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the forward problem admits solutions with 'optimal Sobolev regularity,' but the precise Sobolev indices (e.g., H^s for which s) are not indicated; this should be stated explicitly in the introduction or the well-posedness theorem statement.","section":null},{"comment":"Notation for the magnetic potential A(t,x) and electric potential q(t,x) should be introduced once and used consistently; the current abstract alternates between 'magnetic and electric potentials' and 'time-dependent magnetic and electric potentials' without a clear first definition.","section":null},{"comment":"The partial-data result assumes the potentials are known near the boundary; this assumption should be stated as a numbered hypothesis in the main theorem rather than only in the abstract.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so there are no points requiring point-by-point rebuttal. We will incorporate any minor suggestions during revision.","responses":[],"tokens_in":1113,"tokens_out":58,"duration_ms":8443,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that analyticity on the coefficients lets the Dirichlet-to-Neumann map recover the time-dependent magnetic and electric potentials uniquely, and this holds for full data as well as partial data measured on arbitrarily small boundary sets when the potentials are already known near the boundary. They also prove the forward nonlinear problem is well-posed with optimal Sobolev regularity.\n\nWhat stands out as new is the combination of the nonlinear dynamical setting with a magnetic potential and explicit time dependence, plus the partial-data result under those analyticity assumptions. The partial-data version with small measurement regions is a concrete improvement over earlier full-data statements in this line of work.\n\nThe analyticity assumption is the clearest limitation. It is a standard tool for continuation arguments in inverse problems, but it restricts the result to a narrow class of coefficients and leaves open whether the same uniqueness holds without it. The forward well-posedness claim is load-bearing for the inverse part; if the regularity estimates check out, that part is useful on its own. Nothing in the abstract or claim description points to circular reasoning or mismatched assumptions.\n\nThis is written for people already working on inverse problems for nonlinear Schrödinger operators or magnetic potentials. A reader focused on partial-data uniqueness or time-dependent coefficients will find the statements directly relevant. A broader PDE audience will probably see it as too specialized.\n\nThe paper deserves a serious referee. The claims are specific, the technical pillars are named clearly, and the partial-data extension is worth checking in detail.","headline":"The paper shows uniqueness of time-dependent magnetic and electric potentials from the DN map for a nonlinear dynamical Schrödinger operator under analyticity, plus forward well-posedness at optimal Sobolev regularity, for both full and partial data.","tokens_in":2112,"tokens_out":391,"would_cite":false,"duration_ms":22639,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The Dirichlet-to-Neumann map uniquely determines time-dependent magnetic and electric potentials for a nonlinear dynamical Schrödinger operator under analyticity assumptions.","keywords":["inverse problems","Dirichlet-to-Neumann map","nonlinear Schrödinger equation","magnetic potential","analytic continuation","partial data","uniqueness"],"falsifier":"Finding two distinct sets of analytic potentials that produce identical Dirichlet-to-Neumann maps for the same nonlinear equation would falsify the uniqueness result.","tokens_in":2481,"feed_emoji":"","tokens_out":534,"duration_ms":21231,"temperature":0.7,"pith_summary":"This paper shows that the Dirichlet-to-Neumann map associated with a nonlinear Schrödinger equation containing time-dependent magnetic and electric potentials can recover those potentials uniquely. The uniqueness result applies both when full boundary data is available and when only partial data is measured on small boundary subsets, assuming the potentials are known near the boundary. Analyticity of the potentials is used to extend local information globally via analytic continuation. The work also establishes well-posedness of the forward problem, achieving optimal Sobolev regularity for the solutions.","feed_headline":"DN map recovers time-dependent magnetic and electric potentials","feed_subtitle":"Uniqueness from full and partial boundary data holds under analyticity for the nonlinear operator.","key_machinery":"The Dirichlet-to-Neumann map for the nonlinear dynamical Schrödinger operator with magnetic and electric potentials, which maps boundary inputs to boundary outputs to recover the interior potentials.","core_discovery":"Under suitable analyticity assumptions on the potentials, the Dirichlet-to-Neumann map determines the time-dependent magnetic and electric potentials uniquely from both full and partial boundary measurements for the nonlinear dynamical Schrödinger operator.","pith_inferences":["Relaxing analyticity might require Carleman estimates or other microlocal techniques to retain uniqueness.","The partial-data setup could apply to limited-access settings such as medical or geophysical imaging.","The magnetic potential recovery suggests extensions to vector-potential problems in quantum mechanics."],"forward_implications":["The time-dependent magnetic and electric potentials are uniquely recovered from full Dirichlet-to-Neumann data.","Uniqueness extends to partial data when potentials are known near the boundary and Neumann measurements are taken on arbitrarily small open boundary subsets.","The forward problem admits solutions with optimal Sobolev regularity.","The results apply to both complete and incomplete boundary observations."],"fun_headline_variants":["DN map determines Schrödinger potentials under analyticity","Uniqueness from partial data for magnetic electric potentials","Boundary map recovers nonlinear dynamical Schrödinger potentials","Analytic assumptions yield DN uniqueness for time-dependent potentials"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The potentials satisfy suitable analyticity assumptions that enable uniqueness via analytic continuation.","fun_headline_variants_meta":{"raw":{"variants":["DN map determines Schrödinger potentials under analyticity","Uniqueness from partial data for magnetic electric potentials","Boundary map recovers nonlinear dynamical Schrödinger potentials","Analytic assumptions yield DN uniqueness for time-dependent potentials"]},"model":"grok-4.3","cost_usd":0.004045,"raw_usage":{"total_tokens":1978,"prompt_tokens":505,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":40449500,"prompt_tokens_details":{"text_tokens":505,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1418,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":505,"tokens_out":55,"duration_ms":14676,"temperature":1.0,"reasoning_tokens":1418,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T23:33:40.442395+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding two distinct sets of analytic potentials that produce identical Dirichlet-to-Neumann maps for the same nonlinear equation would falsify the uniqueness result.","supporting_citations":[],"review_version":1}