{"id":"123a07e5-dbbd-4a39-8437-ff1f3887975e","arxiv_id":"2606.21056","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Defines m-order logarithmic Schrödinger operators and proves Taylor expansions for their fractional powers with respect to the order s in (0,1) converging in L^p(R^d).","lead":"The paper defines the m-order logarithm of the Schrödinger operator L_V via spectral measures and extends the definition using the generated semigroup. This is then used to establish Taylor expansions for the fractional powers L_V^s and L_V^{-s} in L^p spaces.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"L^2 spectral definition of log^m L_V does not automatically transfer to L^p-convergence of the Taylor series for L_V^s without explicit L^p-boundedness or kernel estimates","rationale":"The reader's weakest assumption (self-adjointness for the spectral measure) is necessary but not sufficient for the L^p claim; the missing step is the L^p extension and norm convergence, which is the load-bearing gap for the headline result.","tokens_in":1696,"tokens_out":382,"duration_ms":24682,"concrete_test":"Locate the theorem stating the Taylor expansion (likely near §3 or §4) and check whether it contains an L^p-norm estimate or an appeal to L^p-boundedness of log^k L_V; if the proof only cites the L^2 spectral measure and the abstract semigroup extension, recompute the first two terms of the expansion on a test function in L^p∩L^2 for a concrete V (e.g., V=0) and verify whether the L^p remainder vanishes.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires Taylor expansions of L_V^s (s∈(0,1)) converging in L^p norm (1<p<∞). The construction begins with the spectral theorem on a subspace of L^2, then invokes the semigroup {T_t^V} to extend log^m L_V to Lipschitz functions. No indication is given that this extension yields bounded operators on L^p or that the remainder in the Taylor series (in the variable s) can be controlled in L^p norm. For Schrödinger operators the heat semigroup is typically contractive on L^p only under extra assumptions on V (e.g., Kato-class or local integrability yielding Gaussian bounds); without those, the passage from L^2 functional calculus to L^p convergence fails.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.3","summary":"The manuscript defines the m-order logarithmic Schrödinger operator log^m L_V (m ∈ ℕ) for nonnegative potentials V via the spectral measure of the self-adjoint operator L_V on a subspace of L^2(R^d), extends the definition to Lipschitz functions using the semigroup {T_t^V} generated by L_V, and employs these operators to establish Taylor expansions of the fractional powers L_V^s and L_V^{-s} (s ∈ (0,1)) with convergence in L^p(R^d) for 1 < p < ∞.","tokens_in":1875,"tokens_out":473,"duration_ms":17620,"significance":"If the L^p-convergence statements hold, the work would supply a functional-calculus route to fractional powers of Schrödinger operators that is expressed through logarithmic operators; this could be useful for PDE analysis involving such operators. The construction rests on standard spectral and semigroup tools, but its novelty hinges on the rigor of the L^p extension.","major_comments":[{"comment":"Abstract: the central claim asserts L^p-norm convergence of the Taylor series for L_V^s (s ∈ (0,1)). The construction begins with the L^2 spectral theorem and the semigroup extension of log^m L_V; no L^p-boundedness of the resulting operators or explicit control of the Taylor remainder in L^p norm is indicated. For general nonnegative V this transfer from L^2 to L^p typically requires additional assumptions (e.g., Kato-class conditions yielding Gaussian bounds) that are not stated.","section":"Abstract"},{"comment":"Abstract (extension step): the semigroup {T_t^V} is invoked to extend log^m L_V to Lipschitz functions, yet the text gives no indication that these extensions are bounded on L^p(R^d) or that the functional calculus they induce preserves the L^p topology needed for the remainder term in the Taylor expansion of L_V^s to vanish in L^p.","section":"Abstract"}],"minor_comments":[{"comment":"The class of admissible potentials V is described only as 'certain nonnegative potentials'; a precise statement (e.g., local integrability or Kato-class membership) would clarify the setting in which the L^p results are claimed.","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. We agree that the L^p aspects require explicit clarification and will revise the manuscript accordingly to state the necessary assumptions on V and detail the boundedness.","responses":[{"response":"The referee is correct that the abstract omits the conditions on V needed for L^p transfer. The L^2 construction via spectral theorem is complete, but to justify L^p convergence we will add the standing assumption that V belongs to the Kato class (ensuring Gaussian heat kernel bounds). Under this assumption the semigroup is bounded on L^p (1 ≤ p ≤ ∞) and the Taylor remainder can be controlled in L^p norm via the functional calculus. We will revise the abstract, add a remark in the introduction, and include a brief L^p remainder estimate.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the central claim asserts L^p-norm convergence of the Taylor series for L_V^s (s ∈ (0,1)). The construction begins with the L^2 spectral theorem and the semigroup extension of log^m L_V; no L^p-boundedness of the resulting operators or explicit control of the Taylor remainder in L^p norm is indicated. For general nonnegative V this transfer from L^2 to L^p typically requires additional assumptions (e.g., Kato-class conditions yielding Gaussian bounds) that are not stated."},{"response":"We acknowledge the need for explicit L^p boundedness of the extended operators. Under the Kato-class assumption on V (to be stated), the semigroup {T_t^V} extends to a bounded semigroup on L^p and the functional calculus for Lipschitz functions of L_V preserves this boundedness. Consequently the Taylor remainder vanishes in L^p. We will insert a short subsection after the semigroup definition explaining this L^p extension and topology preservation.","revision_made":"yes","referee_comment":"[Abstract] Abstract (extension step): the semigroup {T_t^V} is invoked to extend log^m L_V to Lipschitz functions, yet the text gives no indication that these extensions are bounded on L^p(R^d) or that the functional calculus they induce preserves the L^p topology needed for the remainder term in the Taylor expansion of L_V^s to vanish in L^p."}],"tokens_in":1377,"tokens_out":490,"duration_ms":24190,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is the definition of log^m L_V via the spectral measure on L2, extended by the semigroup, followed by its use to get Taylor expansions for L_V^s and the negative powers with respect to s in (0,1), converging in L^p. That application to the expansions looks like the new piece.\n\nThe setup follows the usual spectral theorem plus semigroup route, which is clean enough for this kind of work. They get credit for spelling out how the higher-order logs give a handle on the fractional-order Taylor series.\n\nThe soft spot is the move to L^p. The construction starts in L2, and the abstract only says \"certain nonnegative potentials.\" Without explicit conditions that guarantee the semigroup is bounded on L^p or that the remainder terms stay controlled there, the convergence claim does not automatically follow. The stress-test note is on target: you typically need Kato-class or similar assumptions on V to get the necessary kernel bounds, and the abstract does not flag that those are in place. If the full proofs supply the estimates, the issue is minor; if not, it is a real gap in the central argument.\n\nThis is aimed at people already working on functional calculus for Schrödinger operators and fractional powers. A reader in that niche could pick up the explicit expansion device.\n\nThe paper shows clear, standard thinking with no internal contradictions, so it is coherent on its own terms.\n\nI would bring it to a reading group only if the group is focused on this corner of PDE theory. I would not cite it in my own work. It deserves peer review so the L^p details can be checked.","headline":"The paper offers a spectral construction of m-order logs for Schrödinger operators and uses them for Taylor expansions of fractional powers in L^p, but the L^p transfer may need more justification on the potential assumptions.","tokens_in":2364,"tokens_out":419,"would_cite":false,"duration_ms":23129,"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":"m-order logarithmic operators yield Taylor expansions for fractional powers of Schrödinger operators in L^p","keywords":["Schrödinger operator","logarithmic operator","fractional powers","Taylor expansion","spectral measure","semigroup","L^p convergence"],"falsifier":"A concrete counter-example would be a nonnegative potential V for which the claimed Taylor series for L_V^s fails to converge in L^p for some s close to 0 or 1 and some p between 1 and ∞.","tokens_in":2612,"feed_emoji":"","tokens_out":654,"duration_ms":20233,"temperature":0.7,"pith_summary":"The paper defines the m-order logarithm of the Schrödinger operator L_V using its spectral measure on a suitable subspace of L^2(R^d) and extends the definition to Lipschitz functions via the semigroup generated by L_V. It then applies these log^m L_V operators to establish Taylor expansions for the fractional powers L_V^s and L_V^{-s} with respect to the order s in (0,1), where the series converge in the L^p(R^d) norm for 1 < p < ∞. A sympathetic reader would care because the result supplies an explicit way to track how these fractional operators vary continuously with their order parameter s.","feed_headline":"Log operators produce Taylor expansions for fractional Schrödinger powers","feed_subtitle":"m-order logs of L_V give series for L_V^s and L_V^{-s} that converge in L^p for s in (0,1)","key_machinery":"The m-order logarithmic operator log^m L_V, constructed from the spectral measure of L_V and extended via its generated semigroup to Lipschitz functions, which is then used to derive the expansions in the fractional order s.","core_discovery":"By defining log^m L_V via the spectral measure of the self-adjoint operator L_V and extending it through the semigroup {T_t^V}, the authors prove Taylor expansions for L_V^s and L_V^{-s} in the variable s ∈ (0,1) that converge in L^p(R^d) for 1 < p < ∞.","pith_inferences":["Truncating the Taylor series could furnish practical approximations to fractional powers for concrete potentials.","The same logarithmic construction might be attempted for other self-adjoint operators to obtain order expansions outside the Schrödinger setting.","If the spectral theory carries over, analogous expansions could be studied on domains with boundary conditions."],"forward_implications":["The expansions hold for both positive and negative fractional powers.","Convergence is obtained in every L^p space with 1 < p < ∞.","The construction applies to a wide class of nonnegative potentials V.","The definition of log^m L_V extends from the spectral measure to Lipschitz functions via the semigroup."],"fun_headline_variants":["m-order logarithmic Schrödinger operators expand fractional powers","Spectral logs of L_V yield Taylor expansions for L_V^s","Log^m L_V enables series expansions of fractional Schrödinger powers","Using logs to Taylor expand L_V^s around order s"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Schrödinger operator L_V must be self-adjoint on a suitable subspace of L^2(R^d) so that a spectral measure exists and can be used to define the logarithmic operators.","fun_headline_variants_meta":{"raw":{"variants":["m-order logarithmic Schrödinger operators expand fractional powers","Spectral logs of L_V yield Taylor expansions for L_V^s","Log^m L_V enables series expansions of fractional Schrödinger powers","Using logs to Taylor expand L_V^s around order s"]},"model":"grok-4.3","cost_usd":0.004791,"raw_usage":{"total_tokens":2339,"prompt_tokens":630,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":47912000,"prompt_tokens_details":{"text_tokens":630,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1645,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":630,"tokens_out":64,"duration_ms":13112,"temperature":1.0,"reasoning_tokens":1645,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T14:05:56.224618+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counter-example would be a nonnegative potential V for which the claimed Taylor series for L_V^s fails to converge in L^p for some s close to 0 or 1 and some p between 1 and ∞.","supporting_citations":[],"review_version":1}