{"id":"309a8506-8578-4255-b478-4fb96cabc884","arxiv_id":"2607.07301","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The leftmost eigenvalues of -h²Δ+iV are asymptotically iE+h^σμ where μ are eigenvalues of model operators at the most degenerate critical points of V, with σ=2α/(α+2).","lead":"This paper proves that for Schrödinger operators with purely imaginary potential, the eigenvalues closest to the imaginary axis are determined by the most degenerate critical points of the potential, with explicit scaling rates. This matters for understanding decay rates in fluid mechanics shear flows and non-selfadjoint spectral theory more broadly.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The parametrix construction and rank argument are rigorous; the main residual risk is the unproven spectral properties of the model operator P_c in the general non-signed case.","rationale":"The reader correctly identifies the non-degeneracy condition (1.5) as the key structural assumption, and the paper itself demonstrates its necessity via Example 1.7. However, (1.5) is a domain assumption clearly stated and justified — it is not a hidden or questionable hypothesis but rather a scope limitation. The more interesting load-bearing issue is the unproven spectral theory of the model operator P_c in the general (non-signed, d≥2) case. Propositions 1.4 and 1.15 are used as inputs to the main theorem but are not fully established for all cases covered by Assumption 1. This is a foundational gap, not an error in the conditional argument. The paper is transparent about it ('We do not know if Λ_c is always an infinite set'), and the applications (shear flows, Morse case, signed potentials) all fall within the verified cases. The parametrix construction (§2.2), the resolvent estimate for Q (Proposition 2.1, proved in detail in §2.4), and the rank argument (§2.3) are all rigorous and well-executed. The reference [8] being 'in preparation' is not load-bearing for the main theorem — it supports Theorem 1.14 which characterizes Λ_c for 1D odd monomials, but the main theorem only requires that Λ_c be well-defined with the stated resolvent properties. Given the transparency about gaps, the rigor of the conditional argument, and the substantial advance over prior work (Coti Zelati–Gallay [16] for Morse case), ACCEPT with HIGH confidence is appropriate. The correctness risk is 'unknown' rather than 'low' precisely because of the model operator gap, but this does not rise to a level requiring a verdict change.","tokens_in":35964,"tokens_out":958,"duration_ms":212968,"concrete_test":"Verify Proposition 1.15(ii) for a specific non-signed, non-separable homogeneous potential in d=2, e.g. V_c(x) = x_1^2 x_2^2 - (x_1^4 + x_2^4)/2 (which satisfies (1.5)). Numerically compute the resolvent ||(P_c - z)^{-1}|| for z near the real axis but away from any eigenvalues, and check whether the polynomial bound in dist(z, σ(P_c)) holds. If it fails, the spectral projector argument in §2.3 needs modification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.5) depends on Proposition 1.4, which asserts that the model operator P_c = -Δ + iV_c has spectrum Λ_c consisting of finite-multiplicity eigenvalues with positive real part tending to +∞. The authors acknowledge: 'We do not know if Λ_c is always an infinite set or at least a non-empty set.' When V_c has a sign, Proposition 1.13 proves Λ_c is infinite via complex dilations. For the 1D monomial case with n odd, Theorem 1.14 (citing [8], 'in preparation') establishes the spectrum. But for a general homogeneous V_c without a sign in dimension d≥2, neither the existence of eigenvalues nor the resolvent estimates of Proposition 1.15 are independently verified — they are stated as properties of P_c and used throughout the parametrix construction (Corollary 1.16 feeds directly into Proposition 2.2 and the rank argument in §2.3). If Λ_c were empty for some admissible V_c, Theorem 1.5 would still be formally true but vacuous near that critical point. The more serious risk is Proposition 1.15(ii): the polynomial resolvent bound ||(P_c - z)^{-1}|| ≤ C + C/dist(z,σ(P_c))^N is asserted without proof for the general case. This estimate is used implicitly in the spectral projector construction via (2.14)-(2.16). However, the paper is transparent about these gaps, the self-adjoint (signed) case covers many applications, and the proof strategy for the main theorem is conditionally correct given the model operator properties. This is a genuine gap in the foundation but not an error in the argument as presented.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper studies the semiclassical Schrödinger operator $P = -h^2 Delta + iV(x)$ on a bounded domain (or torus), where $V$ is a real-valued $C^1$ potential. Under a 'degenerate Morse' assumption (Assumption 1) — requiring that near each critical point $c$ of energy $E$, the potential is approximated by a homogeneous polynomial $V_c$ of degree $alpha_c > 1$ satisfying a non-degeneracy condition on $nabla V_c$ — the authors prove that the leftmost eigenvalues of $P$ near $iE$ are asymptotically given by $iE + h^sigma mu_j + o(h^sigma)$, where $sigma = 2alpha/(alpha+2)$, $alpha$ is the maximal vanishing order at energy $E$, and $mu_j$ are eigenvalues of the model operators $P_c = -Delta + iV_c$ associated to the most degenerate critical points (Theorem 1.5). A resolvent estimate $|(P-z)^{-1}| leq C h^{-sigma}$ away from quasi-eigenvalues is also established. Applications to the associated heat equation and shear flow dissipation are given (Theorem 1.17). The proof proceeds via a parametrix construction combining local model resolvents with an auxiliary elliptic operator $Q = P + G^2$, following the strategy of Coti Zelati–Gallay [16] and Henry [26].","tokens_in":36718,"tokens_out":1528,"duration_ms":248903,"significance":"The paper makes a solid contribution to the spectral theory of non-selfadjoint semiclassical operators. The degenerate Morse setting goes beyond the standard Morse (quadratic) case and allows for $C^1$ potentials with critical points of arbitrary finite order, yielding a continuum of scaling exponents $sigma in (2/3, 2)$. The identification of the optimal spectral gap constant as the infimum of the real part of the model operator spectra, and the eigenmode expansion for the semigroup, are concrete improvements over prior results of [16]. The connection to enhanced dissipation in shear flows provides a natural physical motivation. The authors are transparent about the limitations of the model operator theory in the general (non-signed) case, which is appropriate.","major_comments":[{"comment":"Proposition 1.15(ii) asserts a polynomial resolvent bound $|(mathcal{P}-z)^{-1}| leq C + C/text{dist}(z, sigma(mathcal{P}))^N$ for the model operator $mathcal{P} = -Delta + iV$ in the general case (homogeneous $V$ satisfying (1.5), not necessarily signed). The proof given in Section 3.3 only covers part (i) (the large-$|z|$ regime) in general; for part (ii), the argument at the end of Section 3.3 invokes the discreteness of the spectrum (Proposition 1.12) to obtain (3.28), but the polynomial blow-up rate $N$ is not actually established — it is merely asserted to exist. This estimate is used implicitly in the spectral projector construction (via (2.14)–(2.16) and the contour integrals defining $Pi$ and $Pi_c$). While the rank argument in Section 2.3 may not strictly require the polynomial rate (only that the resolvent is bounded on the contour $partial B(lambda_0, gamma h^sigma)$, which (","section":null},{"comment":"Corollary 1.16 provides), the authors should clarify which properties of $mathcal{P}$ are actually needed for the main theorem and whether Proposition 1.15(ii) is load-bearing. If it is not needed for Theorem 1.5, this should be stated; if it is, the proof gap should be addressed.","section":null}],"minor_comments":[{"comment":"Abstract: 'homogeneous polynomial' should be 'homogeneous function' or 'homogeneous polynomial (when $V in C^infty$)', since Assumption 1 allows $alpha_c notin mathbb{N}$ and $V in C^1$ (see the discussion following (1.6)).","section":null},{"comment":"Example 1.3: The expression $V(x) = sin(4t) r^2 = 4x_1^3 x_2 - x_1 x_2^3$ appears to have a typo — the polynomial $4x_1^3 x_2 - x_1 x_2^3$ does not equal $r^2 sin(4t)$; the correct identity is $r^2 sin(4t) = 4x_1 x_2(x_1^2 - x_2^2)$, i.e., $4x_1^3 x_2 - 4x_1 x_2^3$. Please verify the coefficient 4 on the second term.","section":null},{"comment":"Theorem 1.14 cites [8] as 'in preparation.' If this reference is not yet publicly available, the key asymptotic formula (1.23) and the claim that eigenvalues are simple and real should be briefly justified or the status of [8] clarified, since these properties are used in Example 1.9 and Example 1.10.","section":null},{"comment":"Section 2.4.1, Case 2.2: The construction of the global diffeomorphism $psi$ on $S^{d-1}$ is technically involved. A brief remark summarizing why the non-degeneracy condition (1.5)/(1.7) is essential for this construction (ensuring $d_{theta_0} v_c neq 0$ when $v_c(theta_0) = 0$) would aid readability.","section":null},{"comment":"Equation (1.31): The notation $f_{E,lambda,mu,s}$ has the indices in a different order than the surrounding text (which uses $f_{E,mu}$). Consistency should be fixed.","section":null},{"comment":"The paper would benefit from a brief remark on whether full asymptotic expansions in powers of $h$ are expected when $V in C^infty$, beyond the $o(h^sigma)$ remainder (this is mentioned in passing after Remark 1.8 but could be stated more definitively).","section":null},{"comment":"In the proof of Theorem 1.17, the application of the Gearhart–Prüss inequality via [25, Theorem 1.4] is somewhat terse; a one-sentence explanation of how the resolvent bound (1.37) translates into the semigroup decay would help readers, since this is a key output of the paper.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is well-written and the main theorem is conditionally correct given the model operator properties. The primary concern is the status of Proposition 1.15(ii) in the general (non-signed) case, but as the authors note, the signed case (Proposition 1.13) and the 1D monomial case (Theorem 1.14) cover the main applications. I recommend minor revision with the clarification request on Proposition 1.15(ii)."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"This paper extends the spectral asymptotics for P = -h²Δ + iV(x) from the Morse (non-degenerate) case to degenerate homogeneous critical points in arbitrary dimension. The main result (Theorem 1.5) gives λ_j = iE + h^σ μ_j + o(h^σ) with σ = 2α/(α+2), where α is the maximal vanishing order and μ_j are eigenvalues of model operators P_c = -Δ + iV_c. This genuinely improves on Coti Zelati–Gallay [16] and Henry [26]: they identify the optimal resolvent constant as inf Re σ(P_c), go beyond the vertical line Re z = C*h^σ, and handle C¹ potentials with arbitrary degeneracy orders. The parametrix construction (Section 2.2) is clean, the scaling identity 2−2β−σ = 0 is used consistently, and the rank argument in Section 2.3 is rigorous. The application to shear flows (Theorem 1.17) gives a sharp decay rate with explicit prefactor, which is a nice payoff. The non-degeneracy condition (1.5) on the homogeneous part V_c is load-bearing but reasonable — the paper shows via Example 1.7 that it's necessary, and it ensures isolated critical points with controllable level-set geometry. The real soft spot is the model operator P_c itself. Proposition 1.4 asserts that P_c has finite-multiplicity eigenvalues with positive real part going to +∞, but the authors openly admit they don't know if Λ_c is always non-empty for general (non-signed) V_c in dimension d ≥ 2. When V_c has a sign, Proposition 1.13 handles it via complex dilations. For 1D monomials with n odd, they cite [8] (in preparation). But for a general sign-changing homogeneous V_c in d ≥ 2, neither eigenvalue existence nor the polynomial resolvent bound of Proposition 1.15(ii) is independently verified — and these feed directly into the parametrix and rank arguments. The paper is transparent about this gap, and the argument is conditionally correct given the model operator properties. For applications (shear flows, signed potentials), the foundation is solid. The gap matters for the full generality claimed but doesn't undermine the core contribution. This deserves a serious referee — the strategy is sound, the execution is careful, and the results that are fully justified are already a real advance. The referee should push the authors to either prove the model operator properties in general or clearly delineate which cases are fully established.","headline":"Solid extension of non-selfadjoint semiclassical spectral asymptotics to degenerate critical points; the main theorem is conditionally correct but rests on an incompletely verified model operator.","tokens_in":36767,"tokens_out":626,"would_cite":true,"duration_ms":68046,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Degenerate critical points control spectra of non-self-adjoint Schrödinger operators","keywords":[],"falsifier":"A potential V satisfying the approximation condition (1.6) but violating the non-degeneracy condition (1.5), such as V(x₁,x₂) = x₁² in R², where the model operator -Δ + ix₁² has non-discrete spectrum and the eigenvalue asymptotic of Theorem 1.5 cannot hold.","tokens_in":36182,"feed_emoji":"🌊","tokens_out":1119,"duration_ms":92253,"temperature":0.7,"pith_summary":"For the operator P = -h²Δ + iV(x) on a bounded domain, where V is real-valued and the semiclassical parameter h → 0, the eigenvalues closest to the imaginary axis are determined not by the global shape of V but by the local structure of V at its most degenerate critical points. When V near a critical point c is approximated by a homogeneous polynomial V_c of degree α_c, the leftmost eigenvalues near the energy iE take the form λ = iE + h^σ μ + o(h^σ), where σ = 2α/(α+2), α is the maximal vanishing order among critical points at energy E, and μ ranges over the eigenvalues of the model operator P_c = -Δ + iV_c. The resolvent is bounded by C h^{-σ} away from these quasi-eigenvalues, which is optimal. The result extends from the standard self-adjoint theory (where the max-min principle and quadratic approximations suffice) to the non-self-adjoint setting, where the resolvent is not controlled by distance to the spectrum and pseudospectral effects dominate.","feed_headline":"Degenerate critical points control spectra of non-self-adjoint operators","feed_subtitle":"Leftmost eigenvalues of -h²Δ+iV are governed by local homogeneous models at the most degenerate critical points, with sharp resolvent bounds","key_machinery":"The proof rests on three pillars: (1) a scaling conjugation U_c that relates the semiclassical operator P near a critical point c to the h-independent model operator P_c, extracting the scaling exponent σ = 2α/(α+2); (2) an auxiliary operator Q = P + G², where G is a cutoff localizing near maximally degenerate critical points, which is elliptic where P is not and satisfies the resolvent bound ||(Q-z)^{-1}|| ≤ C h^{-σ}; (3) a parametrix R(z) = Σ φ_c (P_c - z)^{-1} φ_c + ψ (Q-z)^{-1} ψ that patches local model resolvents with the global Q-resolvent via a partition of unity. The rank of the spectral projector of P near each quasi-eigenvalue is shown to equal the sum of ranks of the model Spectr","core_discovery":"The central mechanism is a scaling argument combined with a parametrix construction. Near each maximally degenerate critical point c, the operator P is conjugated by a scaling isometry U_c that zooms in at scale h^β (where β = 2/(α+2)). Under this scaling, P decomposes as h^σ P_c + iE plus lower-order terms, where P_c = -Δ + iV_c is a scale-invariant model operator. The full operator's spectrum near iE is then captured by patching together local resolvents of the model operators P_c at each critical point with a global resolvent estimate for an auxiliary operator Q (which is P plus a confining term G² that makes it elliptic near critical points). The key resolvent estimate for Q uses a Poinc","pith_inferences":[],"forward_implications":["The decay rate of solutions to the heat equation ∂_t u + Pu = 0 is governed by the spectral gap μ₀ = inf Re(Λ), where Λ is the union of model operator spectra, giving the sharp exponential decay rate h^σ μ₀ for the semigroup e^{-tP}.","For shear flows in fluid mechanics governed by ∂_t u + V(x)∂_y u - νΔ u = 0, the enhanced dissipation rate is ν^{α/(α+2)} |k|^{2α/(α+2)}, with the precise prefactor determined by the model operator spectrum.","The framework extends to general complex-valued potentials V with min Re V = 0, where the leftmost eigenvalues should be determined by points where Re V = 0 and Im V' = 0 simultaneously.","The C¹ regularity requirement (rather than C∞) allows treatment of degeneracies of arbitrary order α > 1, yielding the full range of scaling exponents σ ∈ (2/3, 2) for the eigenvalue asymptotics."],"fun_headline_variants":["Imaginary potential spectra governed by local models at degenerate critical points","Semiclassical Schrödinger eigenvalues factor through homogeneous polynomial models","Scaling argument patches local resolvents to capture spectrum of -h²Δ+iV","Most degenerate critical points determine leftmost eigenvalues of non-self-adjoint operato","Shear flow applications from asymptotic eigenvalue localization at degenerate extrema"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The non-degeneracy condition requiring that the homogeneous leading part V_c of the potential has no critical points away from c itself (i.e., ∇V_c(x) ≠ 0 for x ≠ c). This ensures critical points are isolated and the level-set geometry is controllable, but it excludes physically relevant potentials like V = x₁² in two dimensions, where the model operator has non-discrete spectrum and the main theorem fails.","fun_headline_variants_meta":{"raw":{"variants":["Imaginary potential spectra governed by local models at degenerate critical points","Semiclassical Schrödinger eigenvalues factor through homogeneous polynomial models","Scaling argument patches local resolvents to capture spectrum of -h²Δ+iV","Most degenerate critical points determine leftmost eigenvalues of non-self-adjoint operators","Shear flow applications from asymptotic eigenvalue localization at degenerate extrema"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":590,"prompt_tokens":488,"completion_tokens":102,"prompt_tokens_details":null},"tokens_in":488,"tokens_out":102,"duration_ms":21747,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T14:38:59.539906+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A potential V satisfying the approximation condition (1.6) but violating the non-degeneracy condition (1.5), such as V(x₁,x₂) = x₁² in R², where the model operator -Δ + ix₁² has non-discrete spectrum and the eigenvalue asymptotic of Theorem 1.5 cannot hold.","supporting_citations":[],"review_version":1}