{"id":"1d4829e7-25dc-42b3-81a6-698ae93da66d","arxiv_id":"2501.01381","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Schrödinger operators with C^{1,1/2} potentials, the paper proves optimal commutator estimates and explicit rates for local and phase-space Weyl laws, including Hartree minimizers with Coulomb interactions.","lead":"This paper proves quantitative bounds on how quickly Schrödinger eigenprojections converge to classical phase-space limits when the potential is only mildly smooth (C^{1,1/2}). The results give explicit rates for local Weyl laws and extend to interacting fermion systems described by Hartree theory, including Coulomb interactions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central chain rests on an unproved uniformity assertion in the perturbed sharp Weyl law (53); without a proof that Mikkelsen's constant is stable under V → V+E, Lemma 3.1 and all commutator/Weyl conclusions are conditional.","rationale":"The reader's identification of Equation (53) as the weakest assumption is accurate and is confirmed by the manuscript. The paper is a serious and sophisticated contribution: it develops a real method for commutator estimates under C^{1,1/2} potentials, and the downstream arguments in Sections 3–6 are carefully constructed. The specific weakness is not an internal inconsistency but an unproved external uniformity claim about the sharp Weyl law of Mikkelsen. The authors acknowledge that the uniformity is 'not explicitly mentioned in [47]' (Section 3.2) and give a heuristic justification by the nature of the regularization argument. That justification is plausible—the class of potentials is fixed and adding a small constant is a bounded perturbation of the relevant norms on compact sublevel sets—but it is not a proof, and the entire chain of results depends on it. Because the claim is plausible and likely fixable, rejection is not warranted; because the proof is not supplied, full acceptance would be premature. The reader's CONDITIONAL verdict is therefore the right outcome, and my stress-test does not change it. The concrete test I propose is exactly the missing verification: a careful re-derivation of the constants in [47] uniformly in E, or a substitute direct proof of Lemma 3.1. I also note the paper deserves credit for the large amount of surrounding work—Agmon-type estimates, Schatten-norm interpolation, and the Hartree fixed-point argument—which are not in question here.","tokens_in":57546,"tokens_out":2944,"duration_ms":30935,"concrete_test":"Independently re-derive the perturbed sharp Weyl law (53) from [47] for H_E = -ℏ²∆ + V + E with E ∈ [-ε0, ε0]. Specifically, track the C^{1,1/2} norms of the regularized potential V_{ε,E} on Ω_ν(E) = {V+E ≤ ν} and the constants in the pseudo-differential expansion to verify they are bounded uniformly in E. If the regularization scale and all constants can be chosen independent of E, the paper goes through; if the constants diverge as E varies across 0, Lemma 3.1 is unsupported. A complementary check: for a simple one-parameter family of C^{1,1/2} potentials, e.g. V_E = |x|² + E, compare the sharp Weyl remainder bound at E = -ε0, 0, ε0 with the same constant C0 and verify numerically or analytically that the O(ℏ) bound is uniform.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.2, Equation (53), the authors extend Mikkelsen's sharp Weyl law [47, Theorem 1.5] to all E in [-ε0, ε0] with a common constant C0. The justification is a paragraph asserting that although not explicitly in [47], the regularization argument gives constants depending only on the C^{1,1/2} norm of V on Ω_ν = {V ≤ ν}. This is not a proof: adding a constant E changes the sublevel set Ω_ν and the relevant C^{1,1/2} norm, and the pseudo-differential expansion in [47] may not be uniform in E unless this is explicitly verified. Equation (53) is used immediately to prove Lemma 3.1 (the local eigenvalue estimate h^d Tr 1_[a,b](H) ≤ C1(|b-a|+ℏ)), which in turn yields the singular resolvent estimates Lemma 3.2 and ultimately all of Theorem 1.1 and Theorem 1.4. If the uniformity fails, Lemma 3.1 could fail at O(1) level in ℏ, and the commutator bounds—the paper's central contribution—would not follow. The authors themselves flag the step as not explicitly in [47]; this is an unproved premise, not a mere technicality. The same uniformity assumption reappears in the Hartree case at Equation (101), so the gap propagates to the interacting results as well.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes semiclassical Schrödinger operators H = −ℏ²Δ + V with potentials of class C^{1,1/2}_{loc} in dimension d = 3 and establishes Schatten-norm commutator estimates for the spectral projection γ = 1_{H≤0}. These estimates are then used to prove quantitative local and phase-space Weyl laws in L² and L¹, and extended to Hartree minimizers with repulsive singular pair interactions, including the Coulomb case. The technical core is a chain: a sharp Weyl law (from Mikkelsen) yields a local eigenvalue estimate, which yields singular resolvent bounds, which yield the commutator estimates; quantitative convergence of densities and states then follows from coherent-state and variational arguments. The paper is largely self-contained and tracks constants explicitly, but the chain currently rests on a uniformity assertion for the sharp Weyl law under constant potential shifts that is stated but not proved.","tokens_in":57908,"tokens_out":6595,"duration_ms":69689,"significance":"If the missing uniformity is supplied, this is a substantial contribution: it extends commutator estimates from smooth potentials to C^{1,1/2} potentials and to Hartree minimizers with singular interactions, and it provides the first quantitative local and phase-space Weyl law rates in this regularity class. The proof strategy is original in using singular resolvent estimates rather than pseudodifferential calculus, and the authors are careful to record explicit constants and optimal ℏ-scaling in several regimes. However, the main results are conditional on an unproved uniformity assertion in Eq. (53), which is load-bearing: without it, Lemma 3.1 and therefore the commutator and Weyl-law theorems do not follow.","major_comments":[{"comment":"The uniform-in-E perturbed Weyl law (53) is the central load-bearing premise of the paper. It is used immediately to prove Lemma 3.1, which in turn yields the singular resolvent estimates Lemma 3.2 and all of Theorem 1.1 and Theorem 1.4. The justification given in the paragraph after Eq. (52) asserts that Mikkelsen's regularization argument yields constants depending only on the C^{1,1/2} norm on Ω_ν, and that this remains true for V+E with |E| ≤ ε0. This is plausible, but it is not a proof: changing E moves the sublevel set Ω_ν = {V ≤ ν} to {V ≤ ν−E}, and the relevant norms and the pseudodifferential expansion in [47] may depend on E through these sets. The authors should either prove uniformity directly, or state and prove a precise lemma showing that the constant in (53) is the same for all E ∈ [−ε0, ε0]. Without this, the local eigenvalue estimate and all subsequent results are conditional.","section":"Section 3.2, Eq. (53)"},{"comment":"The interacting results inherit exactly the same gap. Equation (101) asserts the uniform perturbed Weyl law for the effective Hamiltonian H_γ = −ℏ²Δ + U + K ∗ ρ_γ, with the same constant C0 for all E ∈ [−ε0, ε0]. This is again justified by the same 'not explicitly mentioned in [47]' argument, but now the potential is ℏ-dependent through ρ_γ, so the uniformity needs to be established for the whole family V_γ. This assumption is used to obtain the local eigenvalue estimate (102), which is needed to control the kernel component q in the fixed-point equation and to run Propositions 3.6 and 3.7 in the proof of Theorem 1.5. Thus Theorems 1.5 and 1.6 are conditional on the same unproved uniformity, and the authors should provide a complete argument before the interacting claims can be accepted.","section":"Section 6.1.1, Eq. (101)"}],"minor_comments":[{"comment":"The paper uses both h and ℏ with h = 2πℏ, but several formulas mix them without explicitly recalling the conversion. For example, Proposition 4.7 is stated for all h, ε > 0, while Theorem 1.4 quotes rates in ℏ; the reader must track the factor 2π throughout. A short notational remark after Eq. (13) would help.","section":"Throughout"},{"comment":"The statement writes ‖ρ_f − ρ_γ‖_{L²(R^{2d})}, but position densities ρ_f and ρ_γ are functions on R^d. This should be R^d; the proof and Theorem 1.4 use the correct domain.","section":"Proposition 4.1"},{"comment":"The phrase 'uniformly in ℏ' in assumptions on V, ∇V, and ∇²V is confusing because V is otherwise fixed and independent of ℏ. If the authors intend a family of potentials depending on ℏ, this should be stated explicitly; otherwise the phrase should be removed or clarified.","section":"Hypotheses (H1), (H3), (H4)"},{"comment":"The condition ‖V‖_{C^{1,α}(Ω_ν)} < C_{Ω_ν} is written as if C_{Ω_ν} were a fixed constant, but the notation suggests it may depend on Ω_ν. It would be clearer to write ‖V‖_{C^{1,α}(Ω_ν)} ≤ C_ν with C_ν depending only on Ω_ν and explicitly on the C^{1,1/2} norm of V on a slightly enlarged set.","section":"Eq. (52)"}],"recommendation":"major_revision","confidential_remarks":"The paper is carefully written and the conditional structure is transparent, which is a virtue. The missing uniformity in Eq. (53) is the single most important issue; it is acknowledged as not explicit in [47], and the given justification is not a proof. I believe this is repairable: the authors should consult Mikkelsen's manuscript and provide a self-contained lemma establishing (53) with full details, including the effect of the shift V → V+E on the regularization and on the compact sublevel sets. The same argument then needs to be adapted for the ℏ-dependent effective potential in Eq. (101). If the uniformity cannot be proved, the main theorems should be restated conditionally. I recommend major revision rather than rejection because the gap is localized and the remainder of the argument is detailed and coherent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers what it advertises: the first commutator estimates for spectral projections of Schrödinger operators with only C^{1,1/2} potentials, plus quantitative local and phase-space Weyl laws, and then extends these to Hartree minimizers with Coulomb interactions. The method is refreshingly concrete—no pseudodifferential calculus, just averaged eigenvalue gaps, resolvent bounds, and coherent-state comparisons. The proofs are detailed and written in a way that makes the main ideas visible. The Hartree application, with explicit log factors in the Coulomb case, is new and should be useful to people studying mean-field fermionic dynamics.\n\nThe soft spot is exactly where the reader's report puts it. Equation (53) asserts that Mikkelsen's optimal Weyl law holds uniformly for V+E with E in a fixed interval, and the justification is a paragraph about the regularization argument rather than a proof. This uniformity is load-bearing: Lemma 3.1, the singular resolvent bound, and then all of the commutator and Weyl-law theorems rest on it. The authors flag the point themselves, which is honest, but it is still an unproved premise. My reading is that the claim is probably true and fixable—the C^{1,1/2} norm of V on the relevant sublevel sets should not change in a problematic way under small constant shifts—but I would want to see the argument written out or, failing that, a direct proof of the local eigenvalue estimate that bypasses the perturbed Weyl law. The same issue carries over to the Hartree setting at equation (101).\n\nI would not call this a fatal flaw. The paper is serious, the statements are precise, and the overall structure is coherent. But the conditional verdict is right: the central claims are likely correct, and they deserve to appear in the literature, but not with this gap left as an assertion. A good referee should ask for a proof or a workaround before acceptance.\n\nWho gets value from this: anyone working on semiclassical limits, quantum mean-field theory, or commutator estimates for non-smooth Hamiltonians. It deserves a serious referee, and I would engage with it in that role. For my own work, I would cite the commutator bounds if I needed them, and I would bring it to a reading group focused on semiclassical analysis.","headline":"Genuinely new commutator and Weyl-law estimates for non-smooth potentials, but the proof chain rests on an unproved uniformity claim that a referee should push to fix.","tokens_in":58346,"tokens_out":1578,"would_cite":true,"duration_ms":18784,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J10","81S30","47B47","47B15","47B10","35P30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For C^{1,1/2} potentials, spectral projections obey ℏ-scale commutator bounds, yielding quantitative Weyl laws.","keywords":["commutator estimates","Schatten norms","local Weyl law","semiclassical limit","Schrodinger operators","non-smooth potentials","Hartree minimizers","Thomas-Fermi density"],"falsifier":"Take a $C^{1,1/2}$ potential that grows like $|x|^{3/2}$ with a bounded oscillatory perturbation of order $|x|^{-1/2}$ (so $\\nabla V$ decays but $\\nabla^2V$ grows), and compute, for small $|E|$ shifts, the sharp Weyl remainder $|h^d \\operatorname{Tr} 1_{H\\le E} - \\int_{|\\xi|^2+V\\le E} dx\\, d\\xi|$. If the $\\hbar$-uniform bound (53) fails for some $E\\in[-\\varepsilon_0,\\varepsilon_0]$, the local eigenvalue estimate (54) fails and with it the commutator bounds of Theorem 1.1 and all quantitative Weyl laws in Theorems 1.4 and 1.6.","tokens_in":57305,"feed_emoji":"⚛","tokens_out":10343,"duration_ms":82329,"temperature":0.7,"pith_summary":"The paper proves that for semiclassical Schrödinger operators in three dimensions with potentials that are only $C^{1,1/2}$ (not smooth), growing to infinity, the spectral projection onto negative energies satisfies commutator bounds in every Schatten norm: $h^d \\operatorname{Tr}(|[x,\\gamma]|^p) \\le C\\hbar$ and $h^d \\operatorname{Tr}(|[\\hbar\\nabla,\\gamma]|^p) \\le C\\hbar$ (with at most logarithmic corrections for low $p$ when the Hessian does not decay). These commutator bounds are the quantum analogue of one derivative of regularity, and they are exactly what is needed to make the local Weyl law quantitative: the position density of the projected states converges to the classical density at rate $\\hbar^{1/3}$ in $L^2$ and $\\hbar^{1/2}$ in $L^1$, with analogous phase-space and trace-class rates. The same analysis extends to minimizers of the Hartree energy with repulsive singular pair potentials, including the Coulomb potential, giving $L^1$ convergence of the Hartree density to the Thomas\\u2013Fermi density at rate $\\hbar^{1/2}|\\ln\\hbar|^{1/4}$. A reader should care because previous quantitative Weyl laws relied on smooth potentials and pseudodifferential techniques, while the potentials that appear in mean-field fermionic models are exactly of this non-smooth type.","feed_headline":"For rough potentials, projections obey ℏ-scale commutator bounds","feed_subtitle":"Sharp convergence rates for quantum-to-classical limits, extended to Coulomb-interacting fermions.","key_machinery":"The load-bearing object is the local eigenvalue estimate $h^d \\operatorname{Tr} 1_{[a,b]}(H) \\le C_1(|b-a|+\\hbar)$, derived from a sharp Weyl law for $C^{1,\\alpha}$ potentials (with $\\alpha\\ge1/2$) together with an asserted uniformity of that law under small constant shifts of the potential. From it the paper builds a singular resolvent estimate $h^d \\operatorname{Tr}(1_{H\\le0}(\\hbar-H)^{-2}) \\le C_2\\hbar$, which is the averaged version of an energy-gap condition: the eigenvalues of $H$ are, on average, separated by gaps of order $\\hbar$. General trace-class and Hilbert\\u2013Schmidt commutator bounds (Propositions 3.3\\u20133.5) then convert this resolvent bound into estimates for $[A,\\gamma]$ when $A=x$, $p=-i\\hbar\\nabla$, or a phase-space translation $\\tau_z$, after controlling the operator norms of $\\gamma A$ and the commutator $[A,H]$ via Agmon-type decay estimates. For the quantitative Weyl law, the second pillar is a coherent-state variational principle: the energy gap between the quantized state and a classical phase-space density controls a sum of positive errors, which are then deconvolved using the Besov regularity just obtained. In the Hartree case, the fixed-point equation $\\gamma=1_{H_\\gamma\\le0}+q$ connects the interacting minimizer to the linear theory, and a priori $L^p$ bounds on the mean-field potential $K*\\rho_\\gamma$ supply the required $C^{1,1/2}$ regularity.","core_discovery":"The central discovery is that the spectral projection $\\gamma=1_{-\\hbar^2\\Delta+V\\le0}$ inherits a uniform semiclassical regularity from a single local eigenvalue estimate, $h^d \\operatorname{Tr} 1_{[a,b]}(H) \\le C(|b-a|+\\hbar)$, even when the potential $V$ is only $C^{1,1/2}_{\\mathrm{loc}}$ with $e^{-\\beta|x|}\\nabla V \\in L^\\infty$. From this estimate the authors derive a singular resolvent bound and then general Schatten-norm commutator bounds, showing that $h^d \\operatorname{Tr}(|[x,\\gamma]|^p) \\le C\\hbar$ and $h^d \\operatorname{Tr}(|[\\hbar\\nabla,\\gamma]|^p) \\le C\\hbar$ for all $p\\in[1,\\infty)$, with $C$ independent of $\\hbar$ (and with $|\\ln\\hbar|$ corrections for the momentum commutator at $p\\le2$ unless $\\nabla^2V$ also decays). These bounds imply that the position density and the Wigner transform of the projected state lie in $\\hbar$-uniform Besov spaces, and they are then converted through a coherent-state variational principle into quantitative local and phase-space Weyl laws with rates $\\hbar^{1/3}$, $\\hbar^{1/2}$, and $\\hbar^{1/4}$. The same machinery is applied to minimizers of the Hartree functional with repulsive singular interactions $K(x)=\\kappa|x|^{-a}$, $a\\in(0,1]$: for $a<1$ the optimal commutator and convergence rates hold, while the Coulomb case $a=1$ carries additional $|\\ln\\hbar|^{1/2}$ or $|\\ln\\hbar|^{1/4}$ factors.","pith_inferences":["The linear commutator estimates are formulated in $d\\ge3$ at the level of Propositions 3.6\\u20133.7, so Theorem 1.1's restriction to $d=3$ appears to be a presentation choice rather than a barrier; if so, the quantitative Weyl-law rates carry over to all $d\\ge3$.","The uniformity of the sharp Weyl law of [47] under constant shifts (inequality (53)) is the one external input the authors do not reprove; a future verification would make the proof self-contained, whereas a counterexample would leave only weaker eigenvalue-counting methods and likely destroy the $\\hbar^{1/2}$ density rate.","Because the proof avoids pseudodifferential calculus, the same local-eigenvalue-estimate-to-commutator pipeline should adapt to Schr\\\"odinger operators on graphs or to discrete Laplacians, where sharp Weyl remainders are known, yielding analogous commutator and quantitative-density bounds.","The $\\sqrt{|\\ln\\hbar|}$ correction in the Coulomb Hartree case is likely not optimal; testing the estimates against exactly solvable radial Coulomb-trap models could reveal whether the logarithm is a proof artifact or intrinsic."],"forward_implications":["The local Weyl law holds quantitatively for non-smooth, growing potentials: $\\|\\rho_\\gamma-\\rho_f\\|_{L^2}\\le C\\hbar^{1/3}$ and $\\|\\rho_\\gamma-\\rho_f\\|_{L^1}\\le C\\hbar^{1/2}$ in $d=3$, with explicit constants.","The Wigner transform $f_\\gamma$ of the spectral projection converges to the classical indicator function in $L^2$ at rate $\\hbar^{1/4}$, and the operator difference $\\gamma-\\rho_f$ converges in trace norm at rate $\\hbar^{1/2}$.","The commutator bounds imply uniform-in-$\\hbar$ $W^{1,1}$ regularity of the position density (up to a logarithmic factor), and uniform Besov regularity $\\|\\gamma\\|_{\\dot B^{1/p}_{p,\\infty}}$ for every $p\\in[1,\\infty]$.","For Hartree minimizers with repulsive singular interactions $K(x)=\\kappa|x|^{-a}$, $a<1$, the optimal $\\hbar$-rates hold; in the Coulomb case $a=1$ the rates carry only $|\\ln\\hbar|^{1/2}$ (trace norm) and $|\\ln\\hbar|^{1/4}$ (density in $L^1$).","Convergence in Wasserstein distances follows: $W_p(\\rho_\\gamma,\\rho_f)\\le C\\hbar^{1/(2p)}$ for $p\\in[1,\\infty)$."],"supporting_citations":[{"why":"Supplies the sharp Weyl law for $C^{1,\\alpha}$ potentials from which the local eigenvalue estimate and all subsequent bounds are derived.","marker":"[47]"},{"why":"Introduced the link between the local eigenvalue estimate and commutator bounds for spectral projections; the paper starts from the same estimate but avoids pseudodifferential techniques.","marker":"[23]"},{"why":"Provides the fixed-point description of Hartree minimizers and the Thomas\\u2013Fermi limit that the interacting part extends quantitatively.","marker":"[48]"},{"why":"Establishes the local Weyl law limit under minimal assumptions, the classical benchmark the quantitative rates refine.","marker":"[24]"},{"why":"Supplies the phase-space shift formula, the isometry between $L^2$ Wigner and $L^2$ Schatten norms, and the quantum Besov norms used to state uniform regularity.","marker":"[38]"},{"why":"Introduces the semiclassical convolution and its Young-type inequalities used in the coherent-state variational argument.","marker":"[57]"},{"why":"The origin of the $p=1$ commutator estimates as a requirement for quantitative derivations of Hartree\\u2013Fock dynamics; Theorem 1.1 verifies this requirement for non-smooth potentials.","marker":"[10]"}],"fun_headline_variants":["Rough potentials still yield sharp semiclassical limits","Quantum-to-classical rates hold for non-smooth potentials","Coulomb repulsion tamed in Hartree minimizer bounds","Commutator bounds extend Weyl laws to rough potentials","ℏ-scale regularity from a single eigenvalue estimate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The local eigenvalue estimate, and therefore every stated commutator bound and Weyl-law rate, rests on the assertion that the sharp Weyl law for $C^{1,1/2}$ potentials remains uniform when the potential is shifted by any small constant $E\\in[-\\varepsilon_0,\\varepsilon_0]$, a uniformity the authors affirm follows from the proof of the Weyl law but do not themselves reproduce.","fun_headline_variants_meta":{"raw":{"variants":["Rough potentials still yield sharp semiclassical limits","Quantum-to-classical rates hold for non-smooth potentials","Coulomb repulsion tamed in Hartree minimizer bounds","Commutator bounds extend Weyl laws to rough potentials","ℏ-scale regularity from a single eigenvalue estimate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000414,"raw_usage":{"total_tokens":2174,"prompt_tokens":1017,"completion_tokens":1157,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":1078}},"tokens_in":633,"tokens_out":1157,"duration_ms":8681,"temperature":1.0,"reasoning_tokens":1078,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:28:10.020294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a $C^{1,1/2}$ potential that grows like $|x|^{3/2}$ with a bounded oscillatory perturbation of order $|x|^{-1/2}$ (so $\\nabla V$ decays but $\\nabla^2V$ grows), and compute, for small $|E|$ shifts, the sharp Weyl remainder $|h^d \\operatorname{Tr} 1_{H\\le E} - \\int_{|\\xi|^2+V\\le E} dx\\, d\\xi|$. If the $\\hbar$-uniform bound (53) fails for some $E\\in[-\\varepsilon_0,\\varepsilon_0]$, the local eigenvalue estimate (54) fails and with it the commutator bounds of Theorem 1.1 and all quantitative Weyl laws in Theorems 1.4 and 1.6.","supporting_citations":[{"cited_title":"Sharp semiclassical spectral asymptotics for Schr\\\"odinger operators with non-smooth potentials","cited_arxiv_id":"2309.12015","evidence_quote":"Supplies the sharp Weyl law for $C^{1,\\alpha}$ potentials from which the local eigenvalue estimate and all subsequent bounds are derived."},{"cited_title":"They are uniformly bounded in ℏ thanks toLp estimates","cited_arxiv_id":null,"evidence_quote":"Introduced the link between the local eigenvalue estimate and commutator bounds for spectral projections; the paper starts from the same estimate but avoids pseudodifferential techniques."},{"cited_title":"Fresta, M","cited_arxiv_id":null,"evidence_quote":"Provides the fixed-point description of Hartree minimizers and the Thomas\\u2013Fermi limit that the interacting part extends quantitatively."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the local Weyl law limit under minimal assumptions, the classical benchmark the quantitative rates refine."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the phase-space shift formula, the isometry between $L^2$ Wigner and $L^2$ Schatten norms, and the quantum Besov norms used to state uniform regularity."},{"cited_title":"Proposition 3.3 then follows forλ ≥ ℏ thanks to Lemma 3.1 and Lemma 3.2","cited_arxiv_id":null,"evidence_quote":"Introduces the semiclassical convolution and its Young-type inequalities used in the coherent-state variational argument."},{"cited_title":"Benedikter, M","cited_arxiv_id":null,"evidence_quote":"The origin of the $p=1$ commutator estimates as a requirement for quantitative derivations of Hartree\\u2013Fock dynamics; Theorem 1.1 verifies this requirement for non-smooth potentials."}],"review_version":1}