{"id":"dd31d858-b32d-4a92-a494-ba4be971e478","arxiv_id":"2606.09358","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Generic compactly supported potentials achieve maximal resonance density for Schrödinger operators in all dimensions, with new proofs and bounds established for even dimensions.","lead":"The paper proves that generic compactly supported potentials make the Schrödinger operator achieve the highest possible density of resonances, with the integrated counting function hitting the optimal asymptotic bound. This completes the picture for even dimensions using complex analysis on Fredholm determinants and may inform scattering models in quantum mechanics.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Even-dimensional Dinh-Vu analogue rests on unverified analytic properties of the Fredholm determinant in the scattering matrix","rationale":"The load-bearing step is exactly the one flagged by the reader: validity of the Fredholm-determinant characterization for direct application of complex-analysis results in even dimensions. No other internal inconsistency is visible from the abstract and claim structure.","tokens_in":1721,"tokens_out":392,"duration_ms":15804,"concrete_test":"Extract the precise statement and proof of the even-dimensional Dinh-Vu analogue (likely the main theorem in the even-dim section). Verify that the Fredholm determinant is shown to satisfy the exact analyticity, order, and growth conditions required by the cited several-complex-variables theorem; recompute the zero-counting asymptotic for the ball potential using the cited uniform Bessel asymptotics and check whether it saturates the claimed sharp bound within 5%.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for even dimensions requires proving an analogue of the Dinh-Vu result: that outside a pluripolar subset of an analytic family of potentials, the integrated resonance counting function achieves the optimal upper bound. This is obtained by characterizing resonances as zeros of Fredholm determinant functions tied to the scattering matrix and then applying results from several complex variables. The abstract notes this constitutes the bulk of the paper and that new results on the determinant are proved. However, even-dimensional scattering matrices involve different phase factors and possible logarithmic terms compared to odd dimensions; if the determinant fails to be holomorphic of finite order in the appropriate half-plane or if its zero-counting does not satisfy the precise growth hypotheses needed for the pluripolar-set argument, the transfer of the Dinh-Vu conclusion does not go through. The paper also claims a sharp upper bound for arbitrary compactly supported potentials and that the ball achieves equality, both of which feed into the generic statement.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for a generic real or complex-valued compactly supported potential, the associated Schrödinger operator achieves maximal resonance density: its integrated resonance counting function attains the optimal asymptotic upper bound. Odd-dimensional cases adapt Dinh-Vu via Christiansen-Hislop; even-dimensional cases form the bulk of the work and include a sharp upper bound for arbitrary compactly supported potentials, equality for the characteristic function of a ball (via uniform Bessel asymptotics), and an even-dimensional analogue of the Dinh-Vu pluripolar-set result. Resonances are characterized as zeros of Fredholm determinants tied to the scattering matrix, permitting application of one- and several-complex-variable techniques.","tokens_in":1935,"tokens_out":536,"duration_ms":18203,"significance":"If the even-dimensional analytic results hold, the paper would establish a substantial extension of resonance-density results to generic potentials in all dimensions, together with new tools (sharp bounds, ball example, determinant properties) that may apply more broadly in scattering theory. The explicit use of complex-analysis methods on the scattering-matrix determinants is a methodological strength.","major_comments":[{"comment":"Even-dimensional Dinh-Vu analogue (bulk of the paper, as described in the abstract): the new results on the Fredholm determinant must explicitly verify holomorphy of finite order in the relevant half-plane and confirm that zero-counting satisfies the precise growth hypotheses needed for the pluripolar-set conclusion; differences in phase factors and possible logarithmic terms relative to odd dimensions are not addressed in the provided description and are load-bearing for transferring the Dinh-Vu argument.","section":"even-dimensional section (bulk of paper)"},{"comment":"Sharp upper bound for arbitrary compactly supported potentials and equality for the ball (abstract and the section proving the ball example): the error estimates arising from uniform Bessel asymptotics must be shown to be strong enough to attain the exact optimal bound without residual logarithmic factors; this feeds directly into the generic statement.","section":"section on ball example and sharp upper bound"}],"minor_comments":[{"comment":"The introduction should state the precise form of the optimal upper bound and the definition of 'maximal resonance density' with a forward reference to the relevant theorem.","section":null},{"comment":"Ensure that all citations to Dinh-Vu, Christiansen-Hislop, and Zworski are complete and that any new determinant results are clearly distinguished from prior work.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments, which help strengthen the exposition of our results on resonance density for Schrödinger operators. We address the major comments point by point below.","responses":[{"response":"The even-dimensional analysis establishes holomorphy of finite order for the relevant Fredholm determinants in the half-plane via the analytic continuation properties of the scattering matrix and associated determinant estimates. The zero-counting function is shown to meet the precise growth hypotheses of the Dinh-Vu theorem (including the necessary order and type conditions), with explicit adjustments for the phase factors that differ from the odd-dimensional case and for any logarithmic contributions arising in the even-dimensional scattering theory. These verifications are load-bearing and appear in the proofs of the new results on the determinants and the pluripolar-set conclusion. To address the referee's concern about explicitness, we will add a clarifying paragraph or subsection summarizing these verifications and their relation to the odd-dimensional case.","revision_made":"yes","referee_comment":"[even-dimensional section (bulk of paper)] Even-dimensional Dinh-Vu analogue (bulk of the paper, as described in the abstract): the new results on the Fredholm determinant must explicitly verify holomorphy of finite order in the relevant half-plane and confirm that zero-counting satisfies the precise growth hypotheses needed for the pluripolar-set conclusion; differences in phase factors and possible logarithmic terms relative to odd dimensions are not addressed in the provided description and are load-bearing for transferring the Dinh-Vu argument."},{"response":"The sharp upper bound for arbitrary compactly supported potentials is obtained from the Fredholm determinant representation, and the ball example uses uniform Bessel asymptotics to achieve equality with the optimal bound. The error terms in these asymptotics are controlled at a rate sufficient to exclude residual logarithmic factors in the integrated counting function (yielding a clean leading-term asymptotic). This is essential for the subsequent generic statement and is verified directly in the ball computation. We will revise the relevant section to display the error estimates more prominently and confirm the absence of logarithmic remainders.","revision_made":"yes","referee_comment":"[section on ball example and sharp upper bound] Sharp upper bound for arbitrary compactly supported potentials and equality for the ball (abstract and the section proving the ball example): the error estimates arising from uniform Bessel asymptotics must be shown to be strong enough to attain the exact optimal bound without residual logarithmic factors; this feeds directly into the generic statement."}],"tokens_in":1423,"tokens_out":529,"duration_ms":19035,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that the paper settles the even-dimensional version of maximal resonance density for generic compactly supported potentials. It establishes a sharp upper bound on the integrated resonance counting function that holds for every such potential, shows that the ball saturates the bound, and gives an even-dimensional analogue of the Dinh-Vu result on complements of pluripolar sets in analytic families.\n\nWhat is actually new is the even-dimensional work. The odd-dimensional case follows from existing results once the Christiansen-Hislop argument is adapted, but the bulk of the paper is the even-dimensional proofs. These include fresh statements about the Fredholm determinants coming from the scattering matrix so that one- and several-complex-variable tools can be applied. The ball example uses uniform Bessel asymptotics in the style of earlier papers.\n\nThe paper does this cleanly enough on the surface. The citation pattern is standard and non-circular, resting on Dinh-Vu and Christiansen-Hislop plus the new determinant results. The characterization of resonances as zeros of those determinants is the usual one in the field.\n\nThe soft spot is the one flagged in the stress test. Even dimensions bring different phase factors and possible logarithmic terms in the scattering matrix, so the new claims about the determinant being holomorphic of finite order and satisfying the right growth conditions are load-bearing. If those properties hold as stated, the pluripolar-set argument transfers and the generic result follows. If the estimates or the order are off, the even-dimensional half does not go through. The abstract says the authors prove what is needed, but the details on error estimates and the precise half-plane behavior would be the first thing a referee should examine.\n\nThis is for readers already working in resonance counting and scattering theory who know the odd-dimensional literature. A specialist in that subfield will find the new bounds and the even-dimensional extension useful. The paper is technically grounded enough and addresses a recognized open question, so it deserves a serious referee rather than a desk reject.","headline":"This paper closes the even-dimensional case for generic maximal resonance density by proving a sharp upper bound, a ball example, and a Dinh-Vu analogue, but the even-dim Fredholm determinant properties are the part that needs the closest check.","tokens_in":2368,"tokens_out":493,"would_cite":false,"duration_ms":13166,"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":"For generic compactly supported potentials, Schrödinger operators achieve maximal resonance density.","keywords":["Schrödinger operators","resonances","resonance counting function","Fredholm determinants","scattering matrix","generic potentials","compact support","complex variables"],"falsifier":"A concrete compactly supported potential, real or complex, for which the integrated resonance counting function stays strictly below the optimal asymptotic upper bound in some dimension.","tokens_in":2625,"feed_emoji":"","tokens_out":552,"duration_ms":23594,"temperature":0.7,"pith_summary":"The paper shows that generic real or complex-valued compactly supported potentials make the associated Schrödinger operator attain the highest possible resonance density. This means the integrated resonance counting function meets the best known asymptotic upper bound. In odd dimensions the result adapts existing work, while even dimensions require new arguments based on complex analysis. A reader cares because this indicates that maximal density is the typical behavior for these operators rather than a special case.","feed_headline":"Generic potentials give Schrödinger operators maximal resonance density","feed_subtitle":"The integrated counting function hits the optimal asymptotic upper bound for almost every compactly supported potential.","key_machinery":"Fredholm determinant functions associated to the scattering matrix, whose zeros correspond to resonances, to which results from one and several complex variables are applied.","core_discovery":"For a generic compactly supported potential the resonances of the Schrödinger operator, identified as zeros of Fredholm determinant functions from the scattering matrix, satisfy that their integrated counting function achieves the optimal asymptotic upper bound.","pith_inferences":["Maximal resonance density may hold for a dense set of potentials in appropriate topologies.","The techniques could apply to other operators where resonances are zeros of analytic functions.","Non-generic potentials might exhibit lower density but can be perturbed to achieve maximal density.","Similar generic results might exist for resonance problems in higher-order or systems of equations."],"forward_implications":["In odd dimensions the result follows from Dinh-Vu after adapting an argument of Christiansen and Hislop.","A sharp upper bound on the integrated resonance counting function holds for any compactly supported potential in even dimensions.","The characteristic function of a ball has a resonance counting function achieving the optimal upper bound.","An even-dimensional version of the Dinh-Vu result holds for complements of pluripolar subsets of analytic families of potentials."],"fun_headline_variants":["Generic potentials maximize Schrödinger resonance density","Schrödinger operators hit maximal resonance density with generic potentials","Maximal resonance density for generic Schrödinger potentials","Schrödinger resonance density achieves optimal bound for generic potentials"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Resonances correspond to the zeros of Fredholm determinant functions related to the scattering matrix.","fun_headline_variants_meta":{"raw":{"variants":["Generic potentials maximize Schrödinger resonance density","Schrödinger operators hit maximal resonance density with generic potentials","Maximal resonance density for generic Schrödinger potentials","Schrödinger resonance density achieves optimal bound for generic potentials"]},"model":"grok-4.3","cost_usd":0.00648,"raw_usage":{"total_tokens":3013,"prompt_tokens":627,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":64799500,"prompt_tokens_details":{"text_tokens":627,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2329,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":627,"tokens_out":57,"duration_ms":19592,"temperature":1.0,"reasoning_tokens":2329,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T14:10:39.748574+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete compactly supported potential, real or complex, for which the integrated resonance counting function stays strictly below the optimal asymptotic upper bound in some dimension.","supporting_citations":[],"review_version":1}