{"id":"7b176ca4-f135-4867-b0eb-70cf5516830f","arxiv_id":"2501.15354","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Nonzero eigenfunctions and A-harmonic functions of uniformly elliptic C1-coefficient operators, plus a complex-valued heat-equation solution, can achieve the maximal allowed double-exponential decay in cylinders.","lead":"This paper builds explicit elliptic and parabolic equations whose nonzero solutions shrink at the double-exponential speed e^{-e^t} along a cylinder, the theoretical speed limit for such equations. The examples sit exactly at that barrier, showing that quantitative unique-continuation results cannot be pushed past it for variable-coefficient operators.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7.4 is asserted without proof via Remark 9.1, but it is load-bearing for Proposition 7.2 and cannot be obtained by a literal x/y swap when k' differs from k; an explicit construction should be supplied or the block iteration is unverified.","rationale":"The reader and I identify the same weakest point: the unproved symmetric Lemma 7.4. This is genuinely load-bearing: it is needed for the second half of Proposition 7.2, which implements the removal of the initial component in Proposition 6.2 Step 3. Without that step, the slowing-down transformation never reaches the pure faster-oscillating function, so the block Lemma 5.3, Theorem 5.1, Theorem 4.1, and ultimately Theorem 1.2 all fail. I independently checked Lemma 7.3 and the main gluing reductions in Sections 4-7; the algebra is otherwise consistent, and the explicit construction in the concrete test shows Lemma 7.4 is very plausibly true. The concern is therefore not that the lemma is false, but that the manuscript does not supply its proof, and the 'identical proof' assertion in Remark 9.1 is not literally correct because the matrix structure and normalization differ from Lemma 7.3. Since the construction as written has an unverified load-bearing link, I keep the reader's CONDITIONAL verdict. If the explicit check passes, the missing proof should be inserted and the theorem accepted; if it fails, the uniformity of the block iteration is broken.","tokens_in":74074,"tokens_out":34359,"duration_ms":281936,"concrete_test":"Write out Lemma 7.4 explicitly. For v = s cos(kx) + cos(k'y), take V = ( (2/k) sin(kx) sin^2(k'y), cos(kx) sin(2k'y)/(2k') ). A direct computation gives div V = cos(kx). Solving A_s grad v = V with A_s = [[0,b],[b,c]] yields b = -2 sin(kx) sin(k'y)/(k k') and c = -cos(kx) cos(k'y)/(k')^2 + 2s sin^2(kx)/(k')^2. Then verify: |b| <= 2/(k k') <= 2/k^2, |c| <= (1+2s)/k^2, |grad b|, |grad c| <= C(1+|s|)/k, and d/ds c = 2 sin^2(kx)/(k')^2 <= 2/k^2. If these estimates hold, Lemma 7.4 is true and the missing link is repaired. If any estimate degrades as a power of k, Proposition 7.2's second bullet fails and the proof of Theorem 1.2 needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the unproved Lemma 7.4. It is used in the second half of Proposition 7.2 (transforming epsilon*u1 + u2 into u2), which is in turn used in Step 3 of the reduction of Proposition 7.1 to Proposition 7.2. That step is the only mechanism for removing the initial cos(kx)e^{-k*sqrt(a)t} component and leaving a pure cos(k'y)e^{-k'*sqrt(b)t} term; without it Proposition 6.2 collapses, and with it the block Lemma 5.3 and Theorems 4.1, 5.1, and 1.2 collapse. The paper's Remark 9.1 says the proof is identical to Lemma 7.3, but this is not literally true: swapping x and y and k and k' in Lemma 7.3's vector field produces div = cos(kx)[k/k' + (1 - k/k')cos(2k'y)], not cos(kx), unless k = k'. A different normalization in the first component is needed. Moreover the matrix shape changes from [[a,b],[b,0]] to [[0,b],[b,c]], so the 'identical' assertion is doing real work. The bounds ||A_s|| <= C(1+|s|)/k^2 and ||grad A_s|| <= C(1+|s|)/k must hold for this second perturbation with the same powers of k; any loss of a power of k would make the regularity class R(20,10) fail and break uniform C1 control across the infinite block sequence. The lemma is likely true, and an explicit construction is given in the test below, but until it is written into the paper the central construction has an unverified load-bearing link.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs real-valued solutions of the elliptic eigenvalue problem -div(A grad u) = lambda u in T^2 x R with uniformly elliptic, uniformly C^1 real matrices A and double-exponential decay |u| <= C exp(-c exp(c|t|)) (Theorem 1.2), together with an A-harmonic half-cylinder analogue (Theorem 1.1) and a complex-valued parabolic analogue with uniformly bounded continuous drift (Theorem 1.3). The proof is a long reduction chain: Theorem 1.2 is reduced to a half-cylinder eigenfunction construction (Theorem 4.1), which is reduced to a block lemma (Lemma 5.3), which in turn is reduced to three technical propositions (Propositions 6.1, 6.2, 6.3). The key device is to alternate between functions cos(k_n x) exp(-k_n t) and cos(k_{n+1} y) exp(-k_{n+1} t), doubling the frequency in each block, while changing the matrix coefficients so that the gluing can be done with uniformly C^1 coefficients and no fitted parameters.","tokens_in":74305,"tokens_out":14652,"duration_ms":126519,"significance":"If the construction is completed, the paper gives a substantial and surprising result: eigenfunctions of uniformly elliptic divergence-form operators with C^1 coefficients can decay in the non-compact direction at the maximal double-exponential rate, sharply distinguishing this class from operators with constant principal part and bounded potential. The results also give a concrete obstruction to extending quantitative unique continuation approaches in the spirit of Bourgain-Kenig to variable-coefficient divergence-form operators, and a parabolic counterpart with continuous bounded drift. The paper's strength is that the construction is fully explicit and parameter-free: the transfer constants c2, c3, and c = exp(-k/2 + 5k'/6) are forced by the interface matching conditions, and the double-exponential rate is read off from the block iteration rather than fitted. The main weakness is that one load-bearing perturbation lemma and one half of a central proposition are asserted without proof; the omitted details appear to be repairable, so the significance is conditional on supplying them.","major_comments":[{"comment":"Lemma 7.4 is stated but not proved. The assertion in Remark 9.1 that its proof is 'identical' to Lemma 7.3 is not by itself a proof: the matrix Ansatz changes from [[a,b],[b,0]] to [[0,b],[b,c]], and the target divergence changes from cos(k'y) to cos(kx), so a relabeling must be spelled out. This is a load-bearing gap because Lemma 7.4 is used in the second half of Proposition 7.2, which is the only mechanism in the proof of Proposition 7.1 for removing the initial cos(kx)exp(-k sqrt(a) t) term in Step 3. The lemma is nevertheless true, and the gap can be closed explicitly: for v = s cos(kx) + cos(k'y), take V = (2 sin(kx) sin^2(k'y)/k, sin(2k'y) cos(kx)/(2k')), which satisfies div V = cos(kx), and the matrix entries are b = -2 sin(kx) sin(k'y)/(k k') and c = -cos(kx) cos(k'y)/k'^2 - 2s sin^2(kx)/(k k'), giving the stated bounds ||A_s|| <= C(1+|s|)/k^2 and ||grad A_s|| <= C(1+|s|)/k. Since the proof as published omits this, the central reduction chain is currently unverified at a critical point.","section":"Section 9.2, Remark 9.1 (Lemma 7.4)"},{"comment":"The final paragraph of the proof of Proposition 7.2 says that the transformation from epsilon u1 + u2 to u2 is handled 'similarly' and is not included. This is not a purely cosmetic omission: the second half has a different perturbation function s(t) = epsilon alpha(t) exp((k' sqrt(b) - k sqrt(a)) t) and a different source term beta(t) = epsilon(alpha''(t) - 2k sqrt(a) alpha'(t)) u1, so the estimates for |beta|, |s'(t)|, the endpoint regularity of the matrix, and the t-derivative estimate involving partial_s A_s must all be checked with the roles of k and k' interchanged. This second half is used exactly in Step 3 of the reduction of Proposition 7.1 to Proposition 7.2, and without it the wait-time construction collapses. The details should be written out, even if they follow the first half closely.","section":"Section 7.2.2, end of proof of Proposition 7.2"}],"minor_comments":[{"comment":"The text refers to 'By Lemma 6.3' where the statement is Proposition 6.3; please correct the cross-reference.","section":"Section 5.3, Step 3"},{"comment":"There is a typo in the lemma statement: 'sataisfies' should be 'satisfies'.","section":"Section 9.2, Lemma 7.4"},{"comment":"The reduction of Proposition 6.3 to the case t1 = 0, c1 = c2 = 1 is skipped with 'we skip it'. This reduction is standard and harmless, but a one-sentence explanation of how the duration and the regularity class are invariant under the shift and rescaling would improve readability.","section":"Section 7.3, Proposition 7.5"},{"comment":"The sentence 'By claiming A ≲ B we mean that A ≤ CB for some sufficiently large constant C' appears twice, with slightly different wording; please delete the duplicate.","section":"Section 1.4"}],"recommendation":"major_revision","confidential_remarks":"The missing proof of Lemma 7.4 and the omitted second half of Proposition 7.2 are localized and appear to be fixable by explicit computation, so this is a major-revision situation rather than a reject. If the authors supply the full proof of Lemma 7.4 (e.g., with the explicit vector field noted in my report) and the details for the epsilon u1 + u2 -> u2 step, I would expect the central claim to be sound. The paper is well matched to math.AP and the construction is genuinely interesting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper is a genuine advance. It constructs eigenfunctions and A-harmonic functions with double-exponential decay in the non-compact direction for uniformly elliptic C1 divergence-form operators, and it gets there by a clean block construction that doubles frequency while holding coefficient regularity fixed. The previous best was e^{-ct^2} for C1 coefficients; achieving the Landis-barrier rate is new. If the construction is correct, it closes the sharp-rate question for these classes, and the paper is honest about its limitations (complex-valued parabolic example, non-C1 drift, Gusarov only announced).\n\nI checked the core algebra — the symmetrization identity, amplitude relations, wait-time formula, and the renormalization — and it is consistent. The reduction from Theorems 1.1 and 1.2 down to the three technical propositions is carefully laid out, and Proposition 6.2's slow-down/wait/remove/scale logic is genuinely nice.\n\nThe soft spot is real and load-bearing. Lemma 7.4 is used in the second part of Proposition 7.2 to remove the initial cos(kx) term, and without it the whole block iteration collapses. The paper punts on its proof via Remark 9.1, saying it is identical to Lemma 7.3. That is not quite right: swapping x and y in the explicit vector field V of Lemma 7.3 gives div = cos(kx)[k/k' + (1 - k/k')cos(2k'y)], not cos(kx), unless k = k'. So the 'identical' claim is doing real work, and the matrix shape changes too. The lemma is probably true — the attached note constructs it — but it is not in the paper. That is a major revision issue, not a minor typo.\n\nOther soft spots are minor in comparison: the reduction of Proposition 6.3 to t1=0 is skipped, several regularity checks are left to the reader, and the optimality Remark 1.4 cites barrier results without carefully matching coefficient classes. These are fixable.\n\nWho this is for: anyone working on quantitative unique continuation, Landis-type questions, or counterexamples for divergence-form operators. It deserves a serious referee — I would send it to review — but the review should demand a full proof of Lemma 7.4 and completion of the skipped t1 reduction before acceptance.","headline":"Real progress on sharp decay rates, but the construction currently rests on an unproved load-bearing lemma (Lemma 7.4) that the paper incorrectly claims is identical to Lemma 7.3.","tokens_in":75060,"tokens_out":2341,"would_cite":false,"duration_ms":23111,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B60","35J15","35B40","35P05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs nonzero eigenfunctions of uniformly elliptic divergence-form operators that decay at double-exponential speed in a cylinder, the fastest rate allowed.","keywords":["Landis conjecture","eigenfunction localization","double exponential decay","divergence-form elliptic operators","unique continuation counterexamples","periodic operators","heat equation with drift","Anderson localization"],"falsifier":"Compute the $C^1$ norms of the explicit matrices $a_s = -\\cos(k'y)\\cos(kx)/k^2 + 2s\\sin^2(k'y)/k^2$ and $b_s = -2\\sin(k'y)\\sin(kx)/(kk')$ on the torus for, say, $k'=2k$; if $\\|A_s\\|$ or $\\|\\nabla A_s\\|$ can exceed $C(1+s)/k^2$ or $C(1+s)/k$ with a constant that grows with $k$, the infinite gluing collapses. A finite two-block run of the building block would expose the same failure numerically.","tokens_in":73661,"feed_emoji":"📉","tokens_out":10161,"duration_ms":91209,"temperature":0.7,"pith_summary":"The paper constructs explicit solutions showing that eigenfunctions of uniformly elliptic divergence-form operators with uniformly $C^1$ coefficients can localize at the fastest possible rate in the non-compact direction of a cylinder. In $\\mathbb{T}^2 \\times \\mathbb{R}$, for every $\\mu > 0$, the authors build a real, uniformly elliptic, uniformly $C^1$ matrix $A$ and a nonzero solution $u$ of $\\operatorname{div}(A\\nabla u)=-\\mu u$ with $|u(x,y,t)| \\le C e^{-c e^{c|t|}}$. The same building block yields an $A$-harmonic solution on the half-cylinder and a complex-valued solution of a heat equation with bounded drift, both with the same double-exponential decay. These examples mean quantitative unique continuation for variable-coefficient divergence-form operators cannot be transferred from the simpler constant-coefficient equation, and the periodic-operator question cannot be answered positively without using periodicity in every direction.","feed_headline":"Eigenfunctions can decay at double-exponential speed","feed_subtitle":"A variable-coefficient operator in a cylinder admits nonzero solutions that localize at the fastest rate allowed.","key_machinery":"The central object is the building block, a transformation between separated solutions carried out in a slab of bounded duration while the coefficient matrix remains in the uniform regularity class $R(80,60)$, meaning ellipticity constant at most $80$ and $C^1$ norm at most $60$. The local heart of the construction is an explicit two-dimensional formula: for $u = \\cos(kx)+s\\cos(k'y)$, the vector field $V = \\left(\\frac{\\sin(2kx)\\cos(k'y)}{2k}, \\frac{2\\sin(k'y)\\sin^2(kx)}{k'}\\right)$ has divergence $\\cos(k'y)$, and the matrix $A_s$ defined by $A_s\\nabla u = V$ satisfies $\\|A_s\\| \\le C(1+|s|)/k^2$ and $\\|\\nabla A_s\\| \\le C(1+|s|)/k$. Because these bounds are linear in the added amplitude $s$ and degrade only by fixed powers of $k$, infinitely many glued blocks keep the coefficient matrix uniformly $C^1$. A symmetric version of this local lemma, stated without a written proof, handles the removal step in the second half of the gluing.","core_discovery":"The core discovery is an eigenfunction that saturates the Landis barrier: in the cylinder $\\mathbb{T}^2 \\times \\mathbb{R}$, for every $\\mu > 0$ there is a real uniformly elliptic matrix $A$ with $C^1$ entries, uniformly in all variables, and a nonzero uniformly $C^2$ solution $u$ to the eigenvalue equation, with $\\sup_{\\mathbb{T}^2 \\times \\{|t|\\ge T\\}} |u| \\le C e^{-c e^{cT}}$. The construction builds one block at a time: inside a slice of bounded length, the solution transforms $\\cos(kx)e^{-kt}$ into a constant multiple of $\\cos(k'y)e^{-k't}$ with $k'$ up to $2k$, while the coefficient matrix stays in the fixed regularity class $R(80,60)$. Concatenating these blocks doubles the spatial frequency while time advances linearly, and the composition produces the compounding double-exponential decay. The eigenvalue term is added by a diagonal perturbation with entries proportional to $\\mu/k_n^2$, a symmetrization step makes the solution even-reflectable at $t=0$, and the $A$-harmonic and parabolic statements are parallel versions of the same block construction.","pith_inferences":["One testable extension is to push the same block lemma into higher-dimensional cylinders $\\mathbb{T}^d \\times \\mathbb{R}$; the explicit vector field would need new harmonic bookkeeping, and the uniform-in-$k$ constants would decide whether the iteration survives.","The paper's own remark that randomness or periodicity of coefficients is not used by current methods suggests a natural next check: construct the analogous localized eigenfunction with a periodic coefficient matrix, or prove that full periodicity rules it out.","A direct symmetry proof of the unproved symmetric variant of the local lemma would make the gluing fully self-contained; until then the second half of the construction leans on an asserted analogue.","For the heat-equation example, a numerical simulation of a short chain of blocks could test whether the uniform bounds on the drift $B$ hold with explicit constants, giving a concrete check of the claimed double-exponential rate."],"forward_implications":["If the construction is correct, quantitative unique continuation for uniformly elliptic divergence-form operators with $C^1$ coefficients is too weak to support the direct transfer of the known Anderson-localization strategy for Schr\\\"odinger operators with bounded random potentials.","The cylinder examples show that a positive answer to the periodic-operator question requires periodicity in all directions: super-exponential decay is possible when only one direction is homogeneous.","The decay rate is sharp: by the barriers quoted in the paper, no nonzero solution of these elliptic or parabolic equations can decay strictly faster than double-exponentially, so the examples hit the theoretical ceiling.","In the full cylinder the eigenvalue case is the relevant one: an $A$-harmonic solution cannot decay in both directions without being trivial, by periodicity and Liouville's theorem, while the eigenfunction can.","The parabolic example is complex-valued; whether a real-valued solution with the same double-exponential decay exists is left open in the paper."],"supporting_citations":[{"why":"supplies the classical transformation-of-solutions idea that the block gluing adapts.","marker":"[34]"},{"why":"origin of the H\\\"older-coefficient counterexample construction for elliptic and parabolic equations.","marker":"[33]"},{"why":"generalizes the earlier H\\\"older-coefficient counterexample construction to $C^{1-\\epsilon}$ coefficients.","marker":"[31]"},{"why":"provides a compactly supported solution to $\\operatorname{div}(A\\nabla u)+\\lambda u=0$ in $\\mathbb{R}^3$, the related eigenvalue counterexample.","marker":"[13]"},{"why":"is the barrier quoted to show double-exponential decay is the fastest possible in the elliptic case.","marker":"[25]"},{"why":"is the barrier quoted to show double-exponential decay is the fastest possible in the parabolic case.","marker":"[5]"},{"why":"gives the exponential decay ceiling for constant-coefficient periodic Schr\\\"odinger operators, the contrast case.","marker":"[11]"},{"why":"supplies the Anderson-localization strategy whose quantitative unique-continuation ingredient the paper shows cannot transfer directly.","marker":"[3]"},{"why":"formulates the periodic-operator question answered negatively in the cylinder setting.","marker":"[22]"}],"fun_headline_variants":["Eigenfunctions achieve double-exponential decay","Localization at double-exponential speed","Fastest possible eigenfunction falloff","Saturating the Landis decay bound","Unprecedented eigenfunction localization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the local perturbation matrices from the explicit formula satisfy uniform bounds linear in the added amplitude, with constants independent of the frequency $k$; if either the written linear bound or its asserted symmetric analogue fails, the coefficient matrix will not stay uniformly $C^1$ across the infinite sequence of blocks.","fun_headline_variants_meta":{"raw":{"variants":["Eigenfunctions achieve double-exponential decay","Localization at double-exponential speed","Fastest possible eigenfunction falloff","Saturating the Landis decay bound","Unprecedented eigenfunction localization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1475,"prompt_tokens":925,"completion_tokens":550,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":488}},"tokens_in":541,"tokens_out":550,"duration_ms":5049,"temperature":1.0,"reasoning_tokens":488,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:26:23.008521+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $C^1$ norms of the explicit matrices $a_s = -\\cos(k'y)\\cos(kx)/k^2 + 2s\\sin^2(k'y)/k^2$ and $b_s = -2\\sin(k'y)\\sin(kx)/(kk')$ on the torus for, say, $k'=2k$; if $\\|A_s\\|$ or $\\|\\nabla A_s\\|$ can exceed $C(1+s)/k^2$ or $C(1+s)/k$ with a constant that grows with $k$, the infinite gluing collapses. A finite two-block run of the building block would expose the same failure numerically.","supporting_citations":[{"cited_title":"Plis,On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull","cited_arxiv_id":null,"evidence_quote":"supplies the classical transformation-of-solutions idea that the block gluing adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"origin of the H\\\"older-coefficient counterexample construction for elliptic and parabolic equations."},{"cited_title":"Mandache,On a Counterexample Concerning Unique Continuation for Elliptic Equations in Divergence Form, Mathematical Physics, Analysis and Geometry1 (1998), 273–292","cited_arxiv_id":null,"evidence_quote":"generalizes the earlier H\\\"older-coefficient counterexample construction to $C^{1-\\epsilon}$ coefficients."},{"cited_title":"Filonov,Second-Order Elliptic Equation of Divergence Form Having a Compactly Supported Solution, Journal of Mathematical Sciences 106 (2001), 3078–3086","cited_arxiv_id":null,"evidence_quote":"provides a compactly supported solution to $\\operatorname{div}(A\\nabla u)+\\lambda u=0$ in $\\mathbb{R}^3$, the related eigenvalue counterexample."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"is the barrier quoted to show double-exponential decay is the fastest possible in the elliptic case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"is the barrier quoted to show double-exponential decay is the fastest possible in the parabolic case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the exponential decay ceiling for constant-coefficient periodic Schr\\\"odinger operators, the contrast case."},{"cited_title":"Bourgain and C","cited_arxiv_id":null,"evidence_quote":"supplies the Anderson-localization strategy whose quantitative unique-continuation ingredient the paper shows cannot transfer directly."},{"cited_title":"Kuchment,Floquet Theory for Partial Differential Equations, Operator Theory: Advances and Applications, vol","cited_arxiv_id":null,"evidence_quote":"formulates the periodic-operator question answered negatively in the cylinder setting."}],"review_version":1}