{"id":"e99c1ad0-f65d-45fc-b9ec-824aafebcf26","arxiv_id":"2412.04406","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In two dimensions, for scaling-critical electromagnetic Schrödinger operators with mean flux outside 1/2 Z (or with symmetric electric potential in the resonant case), the newly defined intertwining operators are L^p-bounded and transfer key estimates from the magnetic-only operator.","lead":"Building on wave-operator ideas, this paper constructs intertwining operators that conjugate a magnetic Schrödinger operator to one with an additional electric potential, and proves these operators are bounded on L^p for a wide class of critical potentials in two dimensions. The payoff is that dispersive, resolvent, and Bochner-Riesz estimates for the simpler operator automatically transfer to the more complicated one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Resonant-case verification of Definition 2.4(2) omits the cross-term quotient estimates that Theorem 2.9 actually uses.","rationale":"The reader's weakest_assumption correctly identifies the proper-perturbation eigenfunction expansion as the load-bearing premise, and Section 5.2's admission about R^2_jk is the most explicit soft spot. My stress-test sharpens that concern: beyond the derivative failure of R^2_jk, the displayed eigenfunction formulas in the resonant cases contain cross terms e_j2 R_j1,s and e_j1 R_j2,s that are not of the scalar form φ_jk = e_jk(1+R_jk) required by Definition 2.4(2). The proof of Lemma 2.8 and the I3 ≲ I2 step in Section 4 rely on that scalar form, so the resonant-case verification has an unstated assumption. This is not a demonstrated counterexample to Theorem 2.13: the quotient R_j1 = R_j1,c + R^2_j1 + cot(jθ)R_j1,s may still satisfy (a)–(d) because R_j1,s vanishes at the zeros of e_j1, and the nonresonant case in Section 5.1 is substantially more explicit. The concern is therefore addressable rather than fatal, matching the reader's conditional verdict. I would not change the verdict: the paper's central 2D claim is credible but rests on a verification that needs either additional estimates or a generalized multiplier lemma for the resonant cases. The proposed check on a concrete symmetric potential would settle whether the missing quotient estimates actually hold.","tokens_in":28869,"tokens_out":20973,"duration_ms":236542,"concrete_test":"Take a concrete symmetric potential, e.g. a(θ) = cos(2θ), with ~A ∈ Z. Using the Section 5.2 formulas, compute the quotient R_j1(θ) = φ_j1(θ)/e_j1(θ) − 1 = R_j1,c(θ) + R^2_j1(θ) + cot(jθ)R_j1,s(θ), and check whether sup_{j≥ℓ,θ} |R_j1(θ)| = O(1/j), sup_{2^L≥ℓ,θ} Σ_{j=2^L}^{2^{L+1}} |D R_j1(θ)| = O(1/ℓ), and the same dyadic estimate for D R'_j1(θ). If any of these fails, Definition 2.4(2) is not satisfied and Theorem 2.9 cannot be invoked in the resonant case as written; if all hold, the missing verification is a fillable gap and the proof can be repaired by adding these quotient estimates, or by proving an explicit matrix-valued analogue of Lemma 2.8 for cross terms of the form e_j2 R_j1,s.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.13 is proved by invoking Theorem 2.9, whose hypothesis is that LA,a is a proper perturbation of LA,0 in the sense of Definition 2.4(2). In the resonant case ~A ∈ Z with a even about π, Section 5.2 states the eigenfunction expansions as φ_j1 = e_j1(1 + R_j1,c + R^2_j1) + e_j2 R_j1,s and φ_j2 = e_j2(1 + R_j2,c + R^2_j2) + e_j1 R_j2,s. Definition 2.4(2) instead requires φ_jk = e_jk(1 + R_jk) with the scalar remainder R_jk satisfying (a)–(d). The text bounds R_j1,c, R_j1,s, R^2_j1 separately, and explicitly concedes that R^2_j1 does not satisfy the derivative estimate, replacing it by a triangle-inequality argument. It never verifies that the actual quotient R_j1 = φ_j1/e_j1 − 1 = R_j1,c + R^2_j1 + cot(jθ)R_j1,s satisfies (a)–(d), especially the dyadic block estimates for D R_j and D R'_j. The same issue appears in Section 5.3. Since the step I3 ≲ I2 in Section 4 is justified by Lemma 2.8 precisely under the scalar-form hypothesis, the proof of Theorem 2.13 in the resonant cases is incomplete as written. This is a gap in the argument, not a refutation of the theorem: the quotient estimates may well hold, but they are neither proved nor supplied as a matrix-valued generalization of Lemma 2.8.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework, in the spirit of Kato wave operators and of the recent work of Miao, Su, and Zheng, for defining spectrally projected intertwining operators W and W* between the electromagnetic Schrödinger operator L_{A,a} and its electric-free counterpart L_{A,0}. The operators are built from angular maps sending e_α to φ_α and radial Hankel-transform compositions, and they satisfy the exact intertwining identity F(L_{A,a}) = W F(L_{A,0}) W*. The central technical result, Theorem 2.9, asserts L^p boundedness of W and W* under an abstract 'proper perturbation' condition on the spherical eigenfunctions. In dimension two, Theorem 2.13 claims a complete result: for a ∈ W^{1,∞}(S^1) and A ∈ W^{1,∞}(S^1,R^2), under the stated conditions on the average magnetic flux ~A and a symmetry condition on a, W and W* are bounded on L^p(R^2) for all 1 < p < ∞. Corollaries then transfer dispersive estimates for the wave propagator, uniform resolvent estimates, and Bochner-Riesz summability from L_{A,0} to L_{A,a}. The proof combines a variable-coefficient discrete multiplier theorem (Lemma 2.3), reduction of the kernel analysis to that of [31] (Proposition 4.1), and asymptotics of the angular eigenfunctions imported from [15,16].","tokens_in":29171,"tokens_out":8646,"duration_ms":84720,"significance":"If the proof is completed, the paper would provide a widely applicable transfer principle for scaling-critical electromagnetic Schrödinger operators: an L^p-bounded functional calculus for the unperturbed operator yields the same for a class of nonconstant spherical perturbations. This goes substantially beyond the constant inverse-square case treated in [31] and is of clear interest for dispersive estimates, uniform resolvent bounds, and Bochner-Riesz summability. The paper is also honest in exposing its dependence on prior eigenfunction asymptotics and in formulating the higher-dimensional situation as a conjecture. However, the 2D resonant cases contain a load-bearing gap in the verification of the 'proper perturbation' hypothesis, so the main theorem as stated is not yet fully proven.","major_comments":[{"comment":"In the resonant cases of Theorem 2.13, the eigenfunction expansions are written as φ_{j1} = e_{j1}(1 + R_{j1,c} + R^2_{j1}) + e_{j2} R_{j1,s} and φ_{j2} = e_{j2}(1 + R_{j2,c} + R^2_{j2}) + e_{j1} R_{j2,s} (and analogously in §5.3). This does not match the scalar form φ_{jk} = e_{jk}(1 + R_{jk}) required by Definition 2.4(2). The actual quotient R_{j1} = φ_{j1}/e_{j1} - 1 equals R_{j1,c} + R^2_{j1} + cot(jθ) R_{j1,s}, and no estimate is given for the cotangent term: bounding R_{j1,s} by O(1/j) does not control the dyadic block sums of D R_{j1} and D R'_{j1} in regions where sin(jθ) is small. The text explicitly concedes in §5.2 that R^2_{j1} does not satisfy the derivative estimate required by Definition 2.4(2). Since the reduction I3 ≲ I2 in the proof of Theorem 2.9 is justified by Lemma 2.8 precisely under the scalar-form hypothesis, and since no matrix-valued generalization of Lemma 2.8 is supplied, the proof of Theorem 2.13 in the resonant cases is incomplete as written.","section":"§5.2 and §5.3; Definition 2.4(2); Lemma 2.8; Theorem 2.9"},{"comment":"Definition 2.4(2) states the smallness conditions (a)–(d) for the averaged remainder R_j(θ) = (1/m_j) Σ_k R_{jk}(θ), whereas Lemma 2.8 and Lemma 2.3 require bounds for the individual variable coefficients R_{jk}(θ) in order to apply the discrete multiplier theorem. The paper does not explain how the averaged estimates imply the individual estimates needed for the multiplier theorem when m_j > 1. In the 2D applications m_j = 2 and the authors prove stronger individual bounds in §5.1, but Theorem 2.9 is stated in full generality, so the gap affects the abstract theorem as stated.","section":"Definition 2.4(2) and Lemma 2.8"},{"comment":"The proof of Lemma 2.3 is only a sketch. The inductive application of the fundamental theorem of calculus is not carried out, and the paper states that 'a precise enumeration could be given with the Faà di Bruno formula' but does not provide it. The assertion that every resulting integral term can be treated as a constant-coefficient multiplier satisfying (2.1)–(2.2) uniformly in the integration variables is essential for the conclusion, yet it is not demonstrated. Since Lemma 2.3 is used in the proofs of Lemma 2.8 and Proposition 4.1, this is a load-bearing technical point that needs a complete proof.","section":"Lemma 2.3"}],"minor_comments":[{"comment":"The phrase 'Assumption 1.11' should read 'Assumption 1.1'.","section":"§5.1"},{"comment":"The reference to Miao–Su–Zheng in the abstract and text is given as 'Tran. Amer. Math. Soc.'; the correct abbreviation is 'Trans. Amer. Math. Soc.'.","section":"References and abstract"},{"comment":"The word 'approzimation' should be 'approximation'.","section":"Reference [37]"},{"comment":"The displayed estimate for the hatted remainder terms contains a duplicated factor: '|\\hat{R}_{jk,s}(θ)|, |\\hat{R}_{jk,s}(θ)||\\hat{R}'_{jk,s}(θ)|, |\\hat{R}'_{jk,s}(θ)|' should list the four quantities |\\hat{R}_{jk,s}|, |\\hat{R}_{jk,c}|, |\\hat{R}'_{jk,s}|, |\\hat{R}'_{jk,c}|.","section":"§5.3"},{"comment":"The abstract has a grammatical slip: 'with a (fixed) magnetic potential an electric potential' should be 'with a (fixed) magnetic potential and an electric potential'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper builds heavily on the authors' own prior work, especially [31] and [15,16], and the 2D theorem depends on importing eigenfunction asymptotics from those papers. The main issue I see is not the overall strategy but the incomplete verification of the scalar remainder hypothesis in the resonant cases; this may be repairable, but it requires either proving the missing quotient estimates or developing a genuinely matrix-valued version of Lemma 2.8. The referee should also weigh whether the averaged-form conditions in Definition 2.4(2) are the right hypotheses for the general theorem as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe bottom line: this paper is worth reading and deserves a referee, but not as-is. The genuinely new item is a 2D theorem (Theorem 2.13) constructing Lp-bounded intertwining operators W, W* between the magnetic Schrödinger operator L_{A,0} and its electromagnetic perturbation L_{A,a}, for a ∈ W^{1,∞}(S^1) and A ∈ W^{1,∞}(S^1,R^2), under a non-resonance condition on the average flux ~A or, in resonant cases, a symmetry condition on a. The corollaries—dispersive estimates, uniform resolvent bounds, Bochner–Riesz summability—follow cleanly once W is bounded. The framework (proper perturbation axiomatics, variable-coefficient discrete multiplier lemma) is a useful abstraction of Miao–Su–Zheng, and the non-resonant case seems solid.\n\nThe soft spot is in the resonant cases, exactly as the stress-test flags. Definition 2.4(2) requires φ_jk = e_jk(1 + R_jk) with a scalar remainder satisfying specific sup and finite-difference bounds. But Section 5.2 (and 5.3) actually prove a matrix expansion: φ_j1 = e_j1(1 + R_{j1,c} + R^2_{j1}) + e_j2 R_{j1,s}, and similarly for φ_j2. The paper explicitly concedes that R^2_{j1} fails the derivative estimate and handles it by triangle inequality. What it never verifies is that the quotient R_j1 = φ_j1/e_j1 − 1 = R_{j1,c} + R^2_{j1} + cot(jθ)R_{j1,s} satisfies Definition 2.4(2)(a)–(d). The cotangent term is not obviously benign; it has poles, and the dyadic block estimates for D R_j and D R'_j are simply not shown. Since step I3 ≲ I2 in Section 4 invokes Lemma 2.8 precisely under the scalar-form hypothesis, the proof of Theorem 2.13 in the resonant cases is incomplete as written.\n\nThis is a gap, not a refutation. The theorem is plausible, and the fix is likely either to verify the quotient estimates (with a clever use of the symmetry of a) or to generalize Lemma 2.8 to matrix-valued remainders. But a referee should push on this before publication. The other soft spots are minor: Lemma 2.3's proof is a sketch with the Faà di Bruno enumeration deferred, and Proposition 4.1 is \"same as [31]\" with details in Appendix A. The abstract also overstates by saying \"complete result\" without mention of the symmetry condition in resonant cases.\n\nI'd send this to a serious referee but request a revision that closes the resonant-case gap or clearly states a weaker theorem.\n\nBest,\n[You]","headline":"New 2D intertwining framework that is elegant but currently has a hole in the resonant-case verification of the proper-perturbation hypothesis.","tokens_in":29773,"tokens_out":3799,"would_cite":false,"duration_ms":33954,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P25","35A23","35Q40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs spectrally projected intertwining operators for electromagnetic Schrödinger Hamiltonians and proves their $L^p$-boundedness in two dimensions, transferring dispersive, resolvent, and Bochner–Riesz estimates.","keywords":["intertwining operator","Stark effect","electromagnetic Schrödinger operator","Lp boundedness","dispersive estimates","uniform resolvent estimates","Bochner-Riesz means","eigenvalue clusters"],"falsifier":"Compute, for an admissible $W^{1,\\infty}$ pair with $\\bar A\\in\\tfrac12\\mathbb{Z}$ and symmetric $a$, the high-energy remainders $R_{jk}$ directly from the Hill equation (5.1); if for some such pair the uniform $O(1/j)$ decay or the difference/derivative bounds of Definition 2.4(2) fail on a positive-density subsequence, the proper-perturbation hypothesis used to pass from the unperturbed to the perturbed eigenbasis would give way and Theorem 2.13 could fail for that pair.","tokens_in":28637,"feed_emoji":"⚛️","tokens_out":11591,"duration_ms":102327,"temperature":0.7,"pith_summary":"This paper tries to show that the Stark–Zeeman splitting of a spherical Schrödinger operator by scaling-critical magnetic and electric potentials does not destroy the wave-operator picture. It defines, cluster by cluster, an intertwining operator $W$ that maps the spectral projections of the electric-free operator $L_{A,0}$ onto those of the perturbed operator $L_{A,a}$, with the identity $F(L_{A,a}) = W F(L_{A,0}) W^*$ for every bounded Borel function $F$. The main discovery is that in dimension two $W$ and $W^*$ are bounded on $L^p(R^2)$ for all $1<p<\\infty$, provided the average magnetic circulation is not a half-integer, or is a half-integer and the electric potential is symmetric about the axis through $\\pi$. If true, this is a black-box transfer: dispersive decay, uniform resolvent estimates, and Bochner–Riesz summability already proved for the electric-free operator automatically hold for the full electromagnetic Hamiltonian. In higher dimensions the same scheme is conditional on a proper-perturbation assumption and recovers the inverse-square-potential example, with zonal-symmetric potentials left as a conjecture.","feed_headline":"Stark-effect Hamiltonians tamed by intertwining wave operators","feed_subtitle":"2D proof transfers dispersive, resolvent, and Bochner–Riesz estimates from the free operator.","key_machinery":"The central object is the spectrally projected intertwining operator $W f = \\sum_{\\alpha} H_{\\tilde\\nu_\\alpha} H_{\\tilde\\mu_\\alpha} f_\\alpha(r)\\,\\varphi_\\alpha(\\theta)$ and its dual $W^*$, where $H_\\nu$ is the Hankel transform of order $\\nu$, $\\mu_\\alpha$ and $\\nu_\\alpha$ are the eigenvalues of the unperturbed and perturbed spherical operators, and $\\varphi_\\alpha$ is the perturbed eigenfunction paired with the unperturbed $e_\\alpha$. The identity $F(L_{A,a})=W F(L_{A,0}) W^*$ is what makes the operator useful. The $L^p$ argument rests on three tools: the new variable-coefficient discrete multiplier theorem (Lemma 2.3), which lets angle-dependent multipliers $C_{jk}(\\theta)$ act on the unperturbed eigenbasis while keeping $L^p$ norms; Mellin-transform bounds for the Hankel composition $H_\\nu H_\\mu$, whose multiplier is a ratio of Gamma functions and whose admissible $p$ range is governed by the first eigenvalues; and a decomposition of the radial kernel into far-from-diagonal pieces, a diagonal singular integral, and error terms. The proper-perturbation hypothesis controls the remainder $R_{jk}$ in $\\varphi_{jk}=e_{jk}(1+R_{jk})$ and is exactly what lets the perturbed eigenbasis inherit the discrete-multiplier property from the unperturbed one.","core_discovery":"The paper's central claim, stated as Theorem 2.13, is that for $a \\in W^{1,\\infty}(S^1)$, $A \\in W^{1,\\infty}(S^1;\\mathbb{R}^2)$ with $A(\\theta)\\cdot\\theta=0$ and $\\lambda_1(A,a)\\ge 0$, the operators $W$ and $W^*$ defined by (1.6)–(1.7) are bounded on $L^p(\\mathbb{R}^2)$ for every $1<p<\\infty$ whenever $\\bar A\\notin \\tfrac12\\mathbb{Z}$, or $\\bar A\\in \\tfrac12\\mathbb{Z}$ and $a(\\pi-\\theta)=a(\\pi+\\theta)$. The proof routes through the general Theorem 2.9, which in any dimension reduces $L^p$-boundedness of the intertwining operators to three ingredients: cluster asymptotics with matching low-frequency bases (Assumption 1.1), a variable-coefficient discrete multiplier theorem for the unperturbed eigenbasis (Lemma 2.3), and the proper-perturbation condition on the eigenfunction remainders (Definition 2.4). In two dimensions each ingredient is verified using the $T$-periodic ODE comparison principle and the eigenfunction asymptotics of [15], and the resulting $L^p$ bounds are then combined with the intertwining identity to transfer the wave-propagator dispersive estimate, the uniform resolvent estimate, and Bochner–Riesz summability from $L_{A,0}$ to $L_{A,a}$.","pith_inferences":["A natural next test is the minimal-regularity question the paper leaves open: if eigenfunction cluster asymptotics survive below $W^{1,\\infty}$, the same proof may push the $L^p$ transfer to rougher electric potentials, and if not, $W^{1,\\infty}$ is close to the natural threshold.","Because $W$ is unitary on $L^2$ but only bounded on $L^p$ for $p\\neq 2$, comparing the transferred dispersive upper bound with lower bounds for the original propagator could show whether the $|t|^{-1/2}$ decay is sharp for magnetic Stark-split systems.","The same scheme should transfer any functional calculus estimate that holds for $L_{A,0}$, such as Strichartz or maximal estimates, without new spectral work, as long as $W$ and $W^*$ are bounded on the relevant spaces.","For zonal potentials in $d\\ge 3$, the missing piece is a verification of Definition 2.4; a direct computation of the high-energy eigenfunction remainders for such potentials would either unlock the conjecture stated in the paper or locate a failure of the proper-perturbation hypothesis."],"forward_implications":["Every admissible two-dimensional $L_{A,a}$ satisfies the wave-propagator bound $\\|\\sin(t\\sqrt{L_{A,a}})/\\sqrt{L_{A,a}}\\,\\phi(\\sqrt{L_{A,a}})f\\|_{L^{p'}} \\le C(1+|t|)^{-\\frac12(\\frac1p-\\frac1{p'})}\\|f\\|_{L^p}$ for $1<p\\le 2$.","Every such operator satisfies the uniform resolvent bound $\\|(L_{A,a}-z)^{-1}f\\|_{L^q} \\le C|z|^{\\frac1p-\\frac1q-1}\\|f\\|_{L^p}$ for $(p,q)$ in the strip (2.3) and all $z\\notin\\mathbb{R}_+$.","Its Bochner–Riesz means $S^\\delta_1(L_{A,a})$ are $L^p$-bounded for every $\\delta>\\delta_c(p,2)$ and $1<p<\\infty$, so $S^\\delta_R(L_{A,a})f\\to f$ in $L^p$.","In dimensions $d\\ge 3$, the conditional theorem already covers the inverse-square potential and would transfer the same family of estimates for any operator that satisfies Assumption 1.1 and Definition 2.4, making higher-dimensional progress a spectral-asymptotics problem.","The $L^2$ intertwining identity is unconditional once the eigenbasis matching is fixed; only the $L^p$-boundedness of the transfer depends on the proper-perturbation hypothesis."],"supporting_citations":[{"why":"Introduces the spectrally projected intertwining operator for constant inverse-square perturbations and supplies the $L^p$ framework and kernel estimates that the present paper extends.","marker":"[31]"},{"why":"Provides the eigenfunction asymptotic formulas (Lemmas 2.1, B.7, B.10) used to verify the proper-perturbation condition in the 2D cases.","marker":"[15]"},{"why":"Establishes eigenvalue-cluster and eigenfunction asymptotics for 2D $W^{1,\\infty}$ potentials, the input for Assumption 1.1 and for the claim that such potentials are proper perturbations.","marker":"[16]"},{"why":"The $T$-periodic spectral comparison principle that pins down which low-frequency eigenfunctions match between $L_{A,0}$ and $L_{A,a}$.","marker":"[10]"},{"why":"The dispersive estimate for $\\sin(t\\sqrt{L_{A,0}})/\\sqrt{L_{A,0}}$ that the intertwining identity transfers to $L_{A,a}$.","marker":"[19]"},{"why":"The uniform resolvent estimate for $L_{A,0}$ that Corollary 2.15 inherits via $W$ and $W^*$.","marker":"[20]"},{"why":"The Bochner–Riesz boundedness for $L_{A,0}$ transferred to $L_{A,a}$ in Corollary 2.16.","marker":"[32]"},{"why":"The spherical-harmonic discrete multiplier theorem that supplies the DMB condition for the unperturbed basis.","marker":"[3]"}],"fun_headline_variants":["Beyond constant Stark: new intertwining operators with L^p bounds","2D proof: intertwining operators tame non-constant potentials","Generalized wave operators handle Stark perturbations in 2D","Stark effect beyond constant shifts: intertwining operators in 2D","Extending intertwining operators to non-constant spherical terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the assumption that each perturbed spherical eigenfunction differs from the unperturbed one by a factor $1+R_{jk}(\\theta)$ with uniformly decaying, well-behaved remainders (the proper-perturbation condition), a property imported from asymptotic formulas in two dimensions and only partially established for one symmetric-case remainder term.","fun_headline_variants_meta":{"raw":{"variants":["Beyond constant Stark: new intertwining operators with L^p bounds","2D proof: intertwining operators tame non-constant potentials","Generalized wave operators handle Stark perturbations in 2D","Stark effect beyond constant shifts: intertwining operators in 2D","Extending intertwining operators to non-constant spherical terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000308,"raw_usage":{"total_tokens":1882,"prompt_tokens":1191,"completion_tokens":691,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":807,"completion_tokens_details":{"reasoning_tokens":604}},"tokens_in":807,"tokens_out":691,"duration_ms":7567,"temperature":1.0,"reasoning_tokens":604,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:25:23.662577+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for an admissible $W^{1,\\infty}$ pair with $\\bar A\\in\\tfrac12\\mathbb{Z}$ and symmetric $a$, the high-energy remainders $R_{jk}$ directly from the Hill equation (5.1); if for some such pair the uniform $O(1/j)$ decay or the difference/derivative bounds of Definition 2.4(2) fail on a positive-density subsequence, the proper-perturbation hypothesis used to pass from the unperturbed to the perturbed eigenbasis would give way and Theorem 2.13 could fail for that pair.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the spectrally projected intertwining operator for constant inverse-square perturbations and supplies the $L^p$ framework and kernel estimates that the present paper extends."},{"cited_title":"Fanelli, V","cited_arxiv_id":null,"evidence_quote":"Provides the eigenfunction asymptotic formulas (Lemmas 2.1, B.7, B.10) used to verify the proper-perturbation condition in the 2D cases."},{"cited_title":"Fanelli, V","cited_arxiv_id":null,"evidence_quote":"Establishes eigenvalue-cluster and eigenfunction asymptotics for 2D $W^{1,\\infty}$ potentials, the input for Assumption 1.1 and for the claim that such potentials are proper perturbations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The $T$-periodic spectral comparison principle that pins down which low-frequency eigenfunctions match between $L_{A,0}$ and $L_{A,a}$."},{"cited_title":"Fanelli, J","cited_arxiv_id":null,"evidence_quote":"The dispersive estimate for $\\sin(t\\sqrt{L_{A,0}})/\\sqrt{L_{A,0}}$ that the intertwining identity transfers to $L_{A,a}$."},{"cited_title":"Fanelli, J","cited_arxiv_id":null,"evidence_quote":"The uniform resolvent estimate for $L_{A,0}$ that Corollary 2.15 inherits via $W$ and $W^*$."},{"cited_title":"Bonami, J","cited_arxiv_id":null,"evidence_quote":"The spherical-harmonic discrete multiplier theorem that supplies the DMB condition for the unperturbed basis."}],"review_version":1}