{"id":"ca3af87f-9572-4a75-ab8d-c1dfbc9f9408","arxiv_id":"2509.18002","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For fractional Schrödinger operators with decaying potentials, global L1 to L∞ dispersive bounds hold in the range (n+1)/4 ≤ α < n/2, under no embedded eigenvalues and zero regularity.","lead":"This paper proves that fractional Schrödinger operators with a decaying potential satisfy the same dispersive decay estimates as the free fractional Laplacian, after projecting away bound states. It fills a gap in a well-studied area by handling the nonlocal, noninteger order case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the conditional spectral hypotheses and the sketched high-energy tail estimates are the only soft spots, but both are standard or fillable.","rationale":"The reader's weakest_assumption points to the spectral hypotheses (no embedded eigenvalues, zero regular). I agree these are assumptions, but they are explicitly stated and are conventional in the literature; a conditional theorem of this kind is a valid mathematical contribution. The more substantive risk is the sketched verification of the high-energy Born-tail derivative bounds in Lemmas 3.3 and 4.3, because those bounds carry the |t|^{-n/(2α)} decay for the tail. I examined the phase-combined form and found that the feared |x||y| factors cancel, making the asserted β thresholds plausible. Thus I do not see a reason to change the reader's CONDITIONAL verdict: the paper should be accepted provided the authors expand the omitted derivative-bound computations. My concrete test targets exactly that expansion.","tokens_in":25047,"tokens_out":40422,"duration_ms":286556,"concrete_test":"Independently derive the j=2 bound (16) in Lemma 4.3 from the expansion (17), using the phase-combined representation ∂_λ^j(e^{-iλ|x|}R_0^+(λ^{2α})(x,·)) = ∂_λ^j(e^{iλ(|x-·|-|x|)}|x-·|^{2α-n}F(λ|x-·|)). Check that no |x||y| factor survives and that the weight ⟨·⟩^{5/2-j+} together with β>n+4 controls every term. If the derivation produces an extra λ power or an unrescued |x||y| factor, the high-energy tail would fail; otherwise the sketch is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reviewing the resolvent machinery, I do not find a load-bearing flaw in the central argument. Theorems 1.1 and 1.2 are explicitly conditional on the absence of embedded eigenvalues and on zero being a regular point of the spectrum; these are standard and, in fact, necessary hypotheses in dispersive-estimate theory, since bound states/threshold resonances destroy the free decay. The low-energy estimates in Propositions 3.4 and 4.4 correctly reduce to invertibility of M±(0), and the difference bounds for M_+^{-1}-M_-^{-1} track the λ^{n-2α} factor from the free resolvent difference. The high-energy tail lemmas (3.3 and 4.3) are the least detailed part of the proof: the derivative bounds (16) are asserted and delegated to 'mimicry' rather than fully derived. However, the key mechanism is visible: the phase is combined as e^{iλ(|x-x1|-|x|)} before differentiation, so phase derivatives produce factors ⟨x1⟩^j instead of |x|^j, and the assumed decay β>n+4 (resp. β>4) supplies enough weight. This makes the sketch credible, and I could not locate an actual countervailing term. The remaining issues are presentation/rigor gaps, not detected errors.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies dispersive estimates for the evolution e^{itH} with H=(-Δ)^α+V on R^n, n≥2, for non-integer α in the range (n+1)/4 ≤ α < n/2 (and in n=2, 3/4≤α<1). Under the explicit assumptions that H has no embedded eigenvalues and that zero is a regular point of the spectrum, the main theorems claim ∥e^{itH}P_ac(H)∥_{L^1→L^∞} ≲ |t|^{-n/(2α)} for n≥3, and ∥e^{itH}H^{1-1/α}P_ac(H)∥_{L^1→L^∞} ≲ |t|^{-1} for n=2. The proof combines detailed pointwise resolvent bounds and expansions (Propositions 2.1–2.2), a quantitative limiting absorption principle (Proposition 2.3), a Born-series decomposition at high energy, and a low-energy analysis via the symmetric resolvent identity with M±(λ)=U+vR_0(λ^{2α})v. Section 5 gives a partial characterization of zero-energy regularity.","tokens_in":25356,"tokens_out":11095,"duration_ms":84134,"significance":"If correct, these are the first global L^1→L^∞ dispersive bounds for perturbed fractional Schrödinger operators in the stated ranges. The resolvent kernel expansions and the limiting absorption principle are potentially useful independent tools. The paper is appropriately conditional: the spectral hypotheses are stated explicitly in the introduction and theorems, and the authors are careful to cite prior work, including their own, for background results. The main weakness is that several load-bearing estimates, especially the high-energy tails, are sketched rather than fully proved. The conditional nature of the main theorems should be kept in mind by readers, since verifying the absence of embedded eigenvalues and zero regularity for the potential class is not addressed.","major_comments":[{"comment":"The derivative bounds (16) are the essential high-energy input for Theorem 1.2, but their proof is only sketched. Formula (17) splits the derivatives, but the required weighted L^2 estimates for ∂_λ^j R_0 for j=1,2 are not stated; Proposition 2.3 gives derivative bounds for R_V, not the spatial weights appearing in (16). The sentence 'the remainder of the proof mimics that of Lemma 3.3' does not show how derivatives falling on the phase e^{-iλ|x|} produce ⟨x1⟩^j factors, nor how the final ⟨x⟩^{1/2-n/(2α)}⟨y⟩^{1/2-n/(2α)} weights arise. Since Proposition 4.1 depends on this estimate, please provide a complete derivation or precise operator-norm statements.","section":"§4.1, Lemma 4.3, Eq. (16)"},{"comment":"In the small-time endpoint case, the proof uses the inequality |2αtλ^{2α-1}+R| ≳ (|t|λ^{2α-1})^{1/2} R^{1/2} without proof. This inequality is not valid near the stationary point λ0=(R/(2α|t|))^{1/(2α-1)} when t and R have opposite signs; the cut-off χ_{λ0} is designed to avoid this, but the support condition must be used explicitly to justify the bound. Without this justification, the claimed |t|^{-2n/(n+1)} decay at the endpoint is not fully established. Please add the missing elementary argument or replace it with a standard non-stationary-phase estimate.","section":"§4.1, Lemma 4.2, endpoint α=(n+1)/4"},{"comment":"The bound (22) is asserted to follow from Proposition 2.1 and Lemma 4.5, but it is a crucial step in the Van der Corput estimate near λ0. The derivation must account for the two cases |x|<|y| and |y|<|x|, the contributions of F± in the resolvent difference, and the way the exponent 1/2-n/(2α) appears after integrating against the weights from M_+^{-1}-M_-^{-1}. As written, the proof jumps from 'we will use that' to the final bound. Please include the kernel-level derivation; the same comment applies to the analogous bound for n=2 in Proposition 3.4.","section":"§4.2, Proposition 4.4, Eq. (22)"}],"minor_comments":[{"comment":"The product in the integrand should be ∏_{i=1}^k V(x_i), not V(x_k); the same typo appears in the line below.","section":"§4.1, Eq. (15)"},{"comment":"The abstract states α∉N, but for n=3 the lower bound is α≥1, so the main theorem includes the integer case α=1; please clarify the wording.","section":"Abstract / Theorem 1.2"},{"comment":"Typo: 'Cauchy-Schwartz' should be 'Cauchy-Schwarz'.","section":"§5, Lemma 5.1"},{"comment":"The paragraph beginning 'The decay on V is necessitated...' is difficult to parse; consider rewriting with explicit norms and indices.","section":"§4.1, Lemma 4.3"}],"recommendation":"major_revision","confidential_remarks":"This is a serious paper whose central claims appear defensible, but the sketched high-energy tail estimates (especially Lemma 4.3) are load-bearing and need to be completed before publication. The conditional spectral assumptions are not a defect, but they deserve more prominence in the abstract. The overlap with the authors' prior work is substantial but appropriately referenced."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the first global dispersive bound for H=(-Δ)^α+V with non-integer α, and the resolvent machinery behind it is the real contribution. The theorems in n≥2 match the free decay rate after projecting off bound states, and the mechanism checks out. The paper does something new: it extends the integer-order results of Goldberg-Green and the free fractional results of Cho-Ozawa-Xia to the perturbed nonlocal setting, and it handles two genuinely fractional phenomena that don't appear for integer α. The high-energy singularity of the resolvent at zero and the low-energy two-power-series expansion (integer powers of λ plus powers of λ^{1/α}, with logs when they collide) are quantified in Propositions 2.1 and 2.2. Proposition 2.3's limiting absorption principle is also useful. Section 5, characterizing zero regularity in terms of distributional solutions, is a nice addition and partially addresses the weakest assumption.\n\nNow the soft spots, in proportion. The high-energy tail lemmas, Lemma 3.3 for n=2 and Lemma 4.3 for n≥3, are the least detailed. The derivative bound (16) in Lemma 4.3 is asserted after a sentence about mimicking the earlier argument, and the derivative estimates in Lemma 3.3 are compressed. The mechanism is visible—by combining the phase as e^{iλ(|x|-|x-x1|)} before differentiating, the problematic |x| factors become ⟨x1⟩^j, and the decay β > n/2α+5/2 supplies enough weight—so I don't see an actual error, but a referee will want those steps written out. The other soft spot is the spectral hypotheses. The theorems are explicitly conditional on the absence of embedded eigenvalues and on zero being a regular point; these assumptions are standard and necessary (bound states at positive energy or threshold resonances genuinely destroy the free decay), and Section 5 clarifies what zero regularity means, but the paper does not prove that the stated potential class satisfies them. That's a limitation of the theorem statement, not a flaw in the argument. The decay rates β>n+4 for n≥3 and β>4 for n=2 are strong but consistent with pointwise resolvent methods.\n\nI agree with the reader's conditional verdict and with the stress-test note: the central argument has no load-bearing flaw. The sketched parts are presentation gaps, not detected errors. The paper is for people working on dispersive estimates, spectral theory of fractional/nonlocal operators, and Strichartz estimates; the resolvent bounds will be reused.\n\nSend it to a serious referee. The core is sound and novel; the tail lemmas need expansion, not a rethink.","headline":"First global dispersive bounds for perturbed fractional Schrödinger operators, with genuinely new resolvent expansions; the main theorems are credible, but two high-energy tail lemmas are sketched and the spectral hypotheses are assumed rather than verified.","tokens_in":25810,"tokens_out":2321,"would_cite":true,"duration_ms":25270,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P25","35R11","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding a decaying potential to a fractional Laplacian does not slow down the absolutely continuous evolution: it inherits the free L1-to-L∞ decay rate, with a smoothing correction in two dimensions.","keywords":["fractional Schrödinger operators","dispersive estimates","resolvent bounds","limiting absorption principle","zero energy regularity","L1-to-L∞ decay","nonlocal Schrödinger equation","Stone's formula"],"falsifier":"A concrete test: choose n=2 or n=3 in the stated α-range, take a compactly supported real-valued V satisfying the decay and spectral assumptions, and compute or simulate ∥e^{itH}P_ac(H)∥_{L1→L∞} at large times; if the decay is slower than |t|^{-n/(2α)} (or |t|^{-1} in n=2), the central claim is false.","tokens_in":24960,"feed_emoji":"⚛️","tokens_out":6836,"duration_ms":51648,"temperature":0.7,"pith_summary":"This paper tries to prove that the long-time spreading of a quantum evolution generated by a fractional Laplacian plus a decaying potential is identical to that of the free fractional Laplacian, after bound states are removed. The central results are conditional on two spectral assumptions: no positive embedded eigenvalues and zero energy being regular. In dimensions n≥3, for fractional orders α between (n+1)/4 and n/2, the bound is ∥e^{itH}P_ac(H)∥_{L1→L∞} ≲ |t|^{-n/(2α)}; in two dimensions, for α in [3/4,1), a smoothed evolution satisfies ∥e^{itH}H^{1−1/α}P_ac(H)∥_{L1→L∞} ≲ |t|^{-1}. These are the first global dispersive bounds for perturbed fractional Schrödinger operators, and they imply Strichartz estimates by standard arguments. The proof works by combining detailed pointwise resolvent estimates with a low-energy expansion and a quantitative limiting absorption principle.","feed_headline":"Perturbed fractional Schrödinger flow decays at free rate","feed_subtitle":"For fractional orders α, the theorem proves |t|^{-n/2α} decay in n≥3 and |t|^{-1} in n=2.","key_machinery":"The main object is the limiting free resolvent R_0^±(λ^{2α}) = ((−Δ)^α − (λ^2 ± i0))^{-1}, whose kernel is represented as e^{iλr} r^{-(n−2α)} F(λr); the paper proves uniform pointwise bounds and a two-term low-energy expansion. Around this, the perturbed resolvent is controlled via the symmetric resolvent identity R_V^± = R_0^± − R_0^± v M_±^{-1} v R_0^±, with M_± = U+vR_0^±v and v=|V|^{1/2}. Stone's formula converts the difference R_V^+−R_V^- into the propagator kernel, and the resolvent bounds supply stationary-phase control of the resulting oscillatory λ-integrals. Invertibility of M_±(0) is exactly what the zero-regularity assumption provides.","core_discovery":"The paper establishes that, for fractional Schrödinger operators H=(−Δ)^α+V with α not an integer, the absolutely continuous part of the evolution decays at exactly the free rate once the hypotheses on the spectrum hold. In dimensions n≥3 and (n+1)/4≤α<n/2, with |V(x)|≲⟨x⟩^{-β}, β>n+4, and with no embedded eigenvalues and zero a regular point, ∥e^{itH}P_ac(H)∥_{L1→L∞}≲|t|^{-n/(2α)}. In two dimensions, for 3/4≤α<1 and β>4, the decay is ∥e^{itH}H^{1−1/α}P_ac(H)∥_{L1→L∞}≲|t|^{-1}. Along the way the paper gives pointwise kernel bounds for all 0<α<n/2 and a quantitative limiting absorption principle for 1/2<α<n/2, and it characterizes zero-energy regularity in terms of distributional solutions of","pith_inferences":["Beyond the paper: the decay condition β>n+4 is likely far from optimal, since integer-order analogues hold near β=2α; a refined low-energy expansion could lower the required decay rate.","Beyond the paper: the restriction α≥(n+1)/4 comes from high-energy resolvent growth, not from zero-energy regularity, so extending to 1/2<α<(n+1)/4 in n≥3 would require a different high-energy mechanism.","Beyond the paper: the free-flow analysis suggests a smoothing-weighted bound e^{itH}H^{n/2(1−1/α)}P_ac(H) with decay |t|^{-n/2} for all dimensions in the range 1/2<α<1; the paper proves only the two-dimensional case.","Beyond the paper: the logarithmic corrections that appear when n=4α in the low-energy expansion hint that endpoint cases may carry log losses or borderline resonance behavior if the method is pushed."],"forward_implications":["In dimensions n≥3, any potential with |V(x)|≲⟨x⟩^{-β}, β>n+4, and with no embedded eigenvalues and zero regular, gives the full free dispersive decay |t|^{-n/(2α)} for e^{itH}P_ac(H).","In two dimensions, for 3/4≤α<1, the same spectral assumptions give |t|^{-1} decay for the smoothed evolution e^{itH}H^{1−1/α}P_ac(H).","The pointwise free-resolvent bounds hold for all 0<α<n/2, and the quantitative limiting absorption principle holds for 1/2<α<n/2, independent of potential assumptions beyond decay.","Standard arguments convert the dispersive bounds into families of Strichartz estimates for fractional Schrödinger equations.","The zero-energy characterization shows that for decaying potentials, zero-energy resonances are possible precisely when 2α<n≤4α, while for n>4α such resonances are expected to be absent."],"fun_headline_variants":["Fractional Schrödinger decay hits free rate","Perturbed fractional flows decay as free","Dispersive bounds for fractional Schrödinger operators","Proof: fractional Schrödinger decay at free rate","Fractional Schrödinger: decay matches free flow"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that H has no positive embedded eigenvalues and that zero is a regular point of the spectrum; the paper assumes this, and the resolvent inversion and Stone-formula step collapse if a potential produces either.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Schrödinger decay hits free rate","Perturbed fractional flows decay as free","Dispersive bounds for fractional Schrödinger operators","Proof: fractional Schrödinger decay at free rate","Fractional Schrödinger: decay matches free flow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1202,"prompt_tokens":720,"completion_tokens":482,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":413}},"tokens_in":464,"tokens_out":482,"duration_ms":4912,"temperature":1.0,"reasoning_tokens":413,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T15:46:48.658807+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test: choose n=2 or n=3 in the stated α-range, take a compactly supported real-valued V satisfying the decay and spectral assumptions, and compute or simulate ∥e^{itH}P_ac(H)∥_{L1→L∞} at large times; if the decay is slower than |t|^{-n/(2α)} (or |t|^{-1} in n=2), the central claim is false.","supporting_citations":[],"review_version":1}