{"id":"8bd62b65-6f7c-40f5-a220-6b132f8bf328","arxiv_id":"2608.07276","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For radial potentials with arbitrary growth, any real-valued solution of Δu=Vu with u(0)≠0 has a sequence of points where |u(x)|≥c e^{-β(|x|)}, with β explicit in terms of the growth bound.","lead":"This paper proves a Landis-type theorem for radial potentials with arbitrary growth: any nonzero real-valued solution of Δu=Vu must have a lower bound e^{-β(r)} along a sequence of points, where β is built from the growth bound. The result gives explicit decay thresholds in every dimension and recovers known polynomial-growth thresholds as a special case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's printed proof is algebraically invalid: equation (2.17) uses r^{-1} where f(r)^{-1} is required, and (2.23) drops a term from (2.22); both are repairable, but the argument must be corrected.","rationale":"The reader's conditional verdict is appropriate. I independently checked Theorem 1.1 and found no flaw; the dominated-convergence step, the bounds on the weighted integrals, and the contradiction beta' <= n < n+1+G(r0) are consistent. For Theorem 1.4, the reader identified the key algebraic problems. My analysis confirms that the pullback in (2.17) is incorrect as written and that (2.23) is missing the c2*int|v| term. These are not merely cosmetic: without the corrected coefficient, the proof does not establish the claimed reduction. However, the errors are repairable under the stated assumptions. The corrected coefficient alpha_corr=f''/f'+(1-n)f'/f is bounded because f'<=1/sqrt(c), f>=f(1)>0 for r>=1, and |f''|<=K|f'|; similarly alpha_corr' is bounded using |f'''/f'|<=K. The omitted integral term only changes the final inequality to beta <= c1 + (c2+1)/beta, which still contradicts for beta large enough. Therefore the mathematical claim is very likely sound, but the printed proof requires correction. This does not change the reader's CONDITIONAL verdict.","tokens_in":7701,"tokens_out":26024,"duration_ms":256071,"concrete_test":"Recompute the pullback of (2.16) under v(r)=w(f(r)) symbolically. If the coefficient is (1-n)f'(r)/f(r), set alpha_corr=f''/f'+(1-n)f'/f and derive the analogues of (2.21)-(2.25). Then verify the stated assumptions imply |alpha_corr|<=c1 and |alpha_corr'|<=c2; if so, and the final inequality reads beta <= c1 + (c2+1)/beta, Theorem 1.4 is valid after correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central Theorem 1.1 is essentially correct: the spherical-mean reduction, the maximum of h=|w|e^beta, the one-sided derivative lower bound, and the cutoff integration all check out. The load-bearing defect is in Theorem 1.4. Equation (2.17) is not the correct pullback of (2.16) under rho=f(r): since w'' is evaluated at rho=f(r), the first-order term is (1-n)f'(r)/f(r) v'(r), not (1-n)r^{-1}f'(r)v'(r). This invalidates the definition of alpha and the bound (2.21) as written. In addition, (2.23) omits the term c2*int|v| present in (2.22), so the displayed inequality is false. These are algebraic errors, not conceptual ones: with the corrected coefficient, the stated assumptions (|f''/f'| and |f'''/f'| bounded, G>=c, f>=f(1)>0, f'<=c^{-1/2}) do imply the needed boundedness of alpha and alpha', and the final contradiction beta <= c1 + (c2+1)/beta still holds for large beta. But as printed, the proof of the Agmon-distance theorem cannot be read as valid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Landis-type decay lower bounds for real-valued solutions of Δu = V u on R^n when V is radial and can grow arbitrarily fast. Theorem 1.1 asserts that if |V| ≤ G for a continuous nonnegative G and u(0) ≠ 0, then |u| is at least c e^{-β(r)} along a sequence of radii tending to infinity, with β(r) = (n+1)r + ∫_0^r G. Theorem 1.4 improves the exponent to β times the Agmon-type distance g(r) = ∫_0^r sqrt(G(s)) ds under the extra assumptions G ∈ C^2, G ≥ c > 0, and boundedness of |f''/f'| and |f'''/f'| for f = g^{-1}. Two corollaries are drawn: a bound A_u ≤ n+2 for bounded radial potentials, and the existence of arbitrarily rapidly growing radial potentials for which the decay threshold remains exponential.","tokens_in":7958,"tokens_out":13239,"duration_ms":120072,"significance":"If the results are correct, Theorem 1.1 provides the first explicit Landis-type threshold for arbitrary continuous radial growth in all dimensions, and the proof is attractively elementary: spherical averaging reduces the problem to a scalar ODE, a maximum argument at r0 gives a lower bound on |w'(r0)|, and a cutoff integration yields the contradiction β'(r0) ≤ n versus β'(r0) = n+1+G(r0). That part of the paper is essentially sound and is a genuine contribution. The Agmon-distance statement in Theorem 1.4 is a natural and valuable strengthening, but its printed proof contains algebraic errors in the reparametrization and in the subsequent inequalities; these are repairable and do not appear to be conceptual, but the proof as printed cannot be accepted without correction. The paper is concise and clearly written in most places, and the main ideas are transparent enough that the corrections should be straightforward.","major_comments":[{"comment":"The pullback of the ODE under v(r) = w(f(r)) is miscomputed. Since w''(f(r)) in (2.16) has coefficient (1-n)ρ^{-1} with ρ = f(r), the first-order term in v'' should be (1-n) f'(r)/f(r) v'(r), not (1-n) r^{-1} f'(r) v'(r). The displayed equation should read v''(r) = [f''(r)/f'(r) + (1-n) f'(r)/f(r)] v'(r) + f'(r)^2 V(f(r)) v(r). Consequently the function α defined after (2.19) is not the coefficient of v', and the bounds (2.21) on α and α' do not follow as stated. With the corrected α, the bounds are still available: |f'(r)| ≤ c^{-1/2}, f is increasing so f(r) ≥ f(1) > 0, and the hypotheses on |f''/f'| and |f'''/f'| then control the additional terms f''/f and (f')^2/f^2. This is a fixable algebraic correction, but it is load-bearing for Theorem 1.4 as written.","section":"§2.4, Eq. (2.17)"},{"comment":"The sign in front of the α v' term in (2.19) is inconsistent with (2.17) and with the integration-by-parts result (2.20). Since v'' = α v' + f'(r)^2 V(f(r)) v, substituting into v'(r0) = -∫ v'' χ - ∫ v' χ' gives -∫ α v' χ, not +∫ α v' χ. With the printed plus sign, integration by parts would produce -α(r0)v(r0) - ∫(α'χ + αχ')v + ∫χ'' v - ∫ f'^2 V v χ, which is not (2.20). Because the later estimates use absolute values, this sign does not affect the final constant after correction, but as printed the derivation is algebraically invalid.","section":"§2.4, Eq. (2.19)"},{"comment":"The passage from (2.22) to (2.23) is false. Sending T to infinity in (2.22) yields |v'(r0)| ≤ c1|v(r0)| + c2∫_{r0}^∞ |v(r)| dr + ∫_{r0}^∞ f'(r)^2 |V(f(r))| |v(r)| dr. The printed (2.23) drops the c2∫|v| term and incorrectly attaches the coefficient c2 to the last integral. Consequently (2.25) should contain a term (c2 + 1) C0 ∫_{r0}^∞ e^{-βr} dr if one uses f'(r)^2 |V(f(r))| ≤ 1, or alternatively the missing term can be absorbed into the generic constant c2. The final contradiction β ≤ c1 + c2/β is preserved, but only after this correction.","section":"§2.4, Eqs. (2.22)–(2.25)"}],"minor_comments":[{"comment":"In the displayed estimate near the end of the proof, the sign of C_f is wrong: since β(r) ≤ (n+1)r + C_f, the factor should be e^{-C_f}, not e^{C_f}. The conclusion is unaffected, but the displayed inequality is incorrect.","section":"§2.3, Corollary 1.3"},{"comment":"The proof should state explicitly that β is chosen large enough at the start. The contradiction β ≤ c1 + c2/β is only a contradiction for β sufficiently large, and the argument establishes the failure of the vanishing assumption for such β; this is a quantifier detail that should be made precise.","section":"§2.4, Theorem 1.4"},{"comment":"The sentence 'boundedness of |f''/f'| then implies boundedness of |f''|' relies on the preceding bound |f'| ≤ c^{-1/2}; it would be clearer to state the product explicitly. Also, the corrected form of α requires the lower bound f(r) ≥ f(1) > 0, which should be noted.","section":"§2.4, after Eq. (2.21)"},{"comment":"There are several typographical errors: 'Corllary' in the heading of §2.3, 'ellitic' in reference [1], 'Nadirashvilli' in reference [9], and 'multiplaction' near the end of §2.4.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The core value of the paper is Theorem 1.1, whose proof I found essentially correct and convincing. Theorem 1.4 is a natural strengthening and the errors in its proof are algebraic and repairable rather than conceptual; I therefore recommend major revision rather than rejection. The author should be asked to correct Eq. (2.17), Eq. (2.19), and the transition (2.22)–(2.25), and to re-verify the constants in the final contradiction. I see no concern about novelty or inappropriate reliance on prior work; the citation to Rossi and Davey is appropriate background."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. Theorem 1.1 is the core result and it's genuinely new: an explicit Landis-type decay threshold for radial potentials with arbitrary continuous growth, in every dimension, which reduces to the known Rossi and Davey thresholds in the bounded and polynomial cases. The spherical-average reduction is clean, and the maximum-point argument, the one-sided derivative bound, and the cutoff integration all check out. There's a harmless sign typo in (2.7), but the overall argument is sound. The corollaries follow directly.\n\nThe soft spot is Theorem 1.4. The stress-test note is right: equation (2.17) is not the correct pullback of (2.16) under rho = f(r). Since w'' is evaluated at f(r), the first-order coefficient in the v-equation is (1-n) f'(r)/f(r), not (1-n) r^{-1} f'(r). And (2.23) drops the c2 integral term that appears in (2.22). These are algebraic slips, not conceptual ones: with the corrected coefficient, the boundedness assumptions on f''/f' and f'''/f' do imply the needed bounds on alpha and alpha', and the final contradiction still goes through for large beta. So the theorem is probably true as stated, but the printed proof cannot be read as valid without correction.\n\nTwo smaller notes. The assumption on f in Theorem 1.4 is imposed separately rather than derived from G; that's a limitation, not an error. The citation pattern looks honest—Rossi, Davey, and the recent counterexample literature are all cited, and I see no inflated claims.\n\nBottom line: the paper does something real and the main result is solid. Theorem 1.4 needs a corrected derivation, but this is referee-fixable in an afternoon. I'd send it out; a careful referee will catch both typos quickly, and the method is worth engaging with. Accept for peer review, conditional on the author fixing the displayed algebra in Section 2.4.","headline":"Theorem 1.1 is genuinely new and essentially correct, but Theorem 1.4's printed proof has two algebraic slips that must be fixed before the paper can be trusted as written.","tokens_in":8517,"tokens_out":2051,"would_cite":true,"duration_ms":18668,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B60","35J10","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every continuous nonnegative radial envelope $G$, any nontrivial real-valued solution of $\\Delta u=Vu$ must have values at least $c e^{-\\beta(r)}$ at a sequence of points going to infinity, with $\\beta(r)=(n+1)r+\\int_0^r G$.","keywords":["unique continuation at infinity","radial potential","arbitrary potential growth","decay threshold","spherical averaging","real-valued solutions","elliptic equation Delta u = V u"],"falsifier":"Check the change of variables in the ODE: substituting $w'(f(r))=(f'(r))^{-1}v'(r)$ into $(1-n)r^{-1}w'(f(r))$ should produce $(1-n)f(r)^{-1}f'(r)v'(r)$, not $(1-n)r^{-1}f'(r)v'(r)$ as written in (2.17); a numerical test with, say, $G(r)=r^2$ will show whether the proof's claimed bound on the coefficient $\\alpha$ is correct after the correction.","tokens_in":7455,"feed_emoji":"📉","tokens_out":9386,"duration_ms":81831,"temperature":0.7,"pith_summary":"This paper asks how fast a nontrivial real-valued solution to $\\Delta u=Vu$ on $\\mathbb{R}^n$ can vanish at infinity when the potential $V$ is radial and may grow without bound. The answer is a decay threshold: for any continuous nonnegative envelope $G$ with $|V|\\le G$, every solution with $u(0)\\ne0$ must have points $x_i$ going to infinity where $|u(x_i)|\\ge c e^{-\\beta(|x_i|)}$, with $\\beta(r)=(n+1)r+\\int_0^r G$. A second theorem improves the exponent to $\\beta\\int_0^r\\sqrt{G}$, matching the sharp polynomial-growth cutoff known for $G(r)=r^N$, under extra regularity of $G$ and its reparametrization. These would be the first explicit decay thresholds valid for arbitrarily growing radial potentials in all dimensions.","feed_headline":"Explicit decay threshold found for any radial potential growth","feed_subtitle":"Spherical averaging turns the PDE into an ODE, yielding exponential lower bounds along a sequence of points going to infinity.","key_machinery":"The argument reduces the PDE to an ordinary differential equation by taking the spherical average $w(r)$ of $u$ over the sphere of radius $r$. The equation for $w$ is $w''=(1-n)r^{-1}w'+V(r)w$, and the proof studies the weighted amplitude $h(r)=|w(r)|e^{\\beta(r)}$. If the desired bound failed, $h$ would tend to zero at infinity and attain a positive maximum at some point $r_0$; at that point the derivative of $w$ has a lower bound from the maximum condition and an upper bound from integrating the ODE against a cutoff, and comparing the two gives the contradiction $\\beta'(r_0)\\le n< n+1+G(r_0)=\\beta'(r_0)$. For the sharper exponent, the change of variable $r=f(s)$, $f=g^{-1}$, converts the weight $g(r)$ into a linear weight $\\beta s$, and the assumed bounds on $f''/f'$ and $f'''/f'$ make the coefficients of the transformed ODE bounded, so the same maximum-point argument applies.","core_discovery":"The paper's central claim is that a nontrivial real-valued solution of $\\Delta u=Vu$ with radial $V$ cannot decay faster than a computable exponential envelope given only the growth of $V$. Theorem 1.1 proves that if $|V|\\le G$ for continuous $G\\ge0$, then for every solution with $u(0)\\ne0$ there are points $x_i\\to\\infty$ with $|u(x_i)|\\ge c e^{-\\beta(|x_i|)}$, where $\\beta(r)=(n+1)r+\\int_0^r G$. Theorem 1.4 proves, under $G\\in C^2$, $G\\ge c>0$, and boundedness of $|f''/f'|$ and $|f'''/f'|$ for $f=g^{-1}$, $g=\\int_0^r\\sqrt G$, the stronger bound $|u(x_j)|\\ge c e^{-\\beta g(|x_j|)}$ with a constant $\\beta>0$ depending only on $G$; this exponent is proportional to the distance in the metric $ds=\\sqrt G\\,dr$ associated with the potential.","pith_inferences":["A natural next step is to test numerically whether the maximum-point contradiction survives when $G$ is merely continuous and oscillates rapidly; the proof of Theorem 1.1 suggests it should, since only the integral of $G$ enters.","The role of real-valuedness points to a genuine obstruction: for complex-valued solutions the same argument fails, and known counterexamples show a slower decay threshold, so the separation between real and complex cases is not an artifact of the proof.","If the apparent factor error in the reparametrized ODE is corrected, the sharper exponent may hold under weaker regularity than the assumed boundedness of $f''/f'$ and $f'''/f'$, possibly extending the result to more rapidly growing potentials such as exponentials."],"forward_implications":["For bounded radial potentials, the first theorem gives $A_u:=\\liminf_{|x|\\to\\infty}\\log(1/|u(x)|)/|x|\\le n+2$, so no nontrivial solution can decay like $e^{-c|x|}$ with $c>n+2$.","A radial potential can be constructed whose values grow arbitrarily fast on integer spheres yet every nontrivial solution still has a purely exponential lower bound $e^{-(n+1)|x_i|}$ along a sequence, because the integral of the envelope can be made finite.","For polynomial growth $G(r)=r^N$, the sharp exponent of Theorem 1.4 becomes proportional to $r^{N/2+1}$, reproducing the known threshold for that class and showing that the associated geodesic exponent is the right rate.","The theorem gives a quantitative unique-continuation-at-infinity statement in every dimension for radial potentials, complementing known counterexamples for nonradial potentials in high dimensions."],"supporting_citations":[{"why":"provides the sharp decay threshold for polynomial-growth potentials that Theorem 1.4 matches","marker":"[3]"},{"why":"shows counterexamples for nonradial potentials in dimensions three and higher, motivating the radial restriction","marker":"[4]"},{"why":"gives the prior quantitative lower bound for bounded potentials in the plane that Theorem 1.1 extends and simplifies","marker":"[9]"},{"why":"establishes the qualitative radial statement in all dimensions that this paper makes quantitative","marker":"[10]"},{"why":"supplies the distance associated with the metric $ds=\\sqrt G\\,dr$ used to interpret the exponent in Theorem 1.4","marker":"[1]"}],"fun_headline_variants":["Decay threshold computable for any radial potential","Landis theorem extends to arbitrary radial growth","No solution decays faster than computable envelope","Radial potentials: explicit lower bounds on decay","Computable decay exponent for any radial potential"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharper theorem assumes, without deriving it from the potential $G$, that the inverse function $f$ of $g(r)=\\int_0^r\\sqrt G$ has bounded ratios $|f''/f'|$ and $|f'''/f'|$; if those bounds fail or if the written coefficient error in the transformed equation is not repaired, the contradiction argument does not go through.","fun_headline_variants_meta":{"raw":{"variants":["Decay threshold computable for any radial potential","Landis theorem extends to arbitrary radial growth","No solution decays faster than computable envelope","Radial potentials: explicit lower bounds on decay","Computable decay exponent for any radial potential"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000398,"raw_usage":{"total_tokens":2046,"prompt_tokens":872,"completion_tokens":1174,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":1106}},"tokens_in":488,"tokens_out":1174,"duration_ms":9100,"temperature":1.0,"reasoning_tokens":1106,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T11:11:39.982838+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the change of variables in the ODE: substituting $w'(f(r))=(f'(r))^{-1}v'(r)$ into $(1-n)r^{-1}w'(f(r))$ should produce $(1-n)f(r)^{-1}f'(r)v'(r)$, not $(1-n)r^{-1}f'(r)v'(r)$ as written in (2.17); a numerical test with, say, $G(r)=r^2$ will show whether the proof's claimed bound on the coefficient $\\alpha$ is correct after the correction.","supporting_citations":[{"cited_title":"Davey, On Landis’ Conjecture in the Plane for Potentials with Growth , Viet","cited_arxiv_id":null,"evidence_quote":"provides the sharp decay threshold for polynomial-growth potentials that Theorem 1.4 matches"},{"cited_title":"Counterexamples to the Landis conjecture in dimensions three and higher","cited_arxiv_id":"2608.00802","evidence_quote":"shows counterexamples for nonradial potentials in dimensions three and higher, motivating the radial restriction"},{"cited_title":"Logunov, E","cited_arxiv_id":null,"evidence_quote":"gives the prior quantitative lower bound for bounded potentials in the plane that Theorem 1.1 extends and simplifies"},{"cited_title":"Rossi, The Landis Conjecture with Sharp Rate of Decay , Indiana Univ","cited_arxiv_id":null,"evidence_quote":"establishes the qualitative radial statement in all dimensions that this paper makes quantitative"},{"cited_title":"Agmon, On exponential decay of solutions of second order ellitic questions in bounded domains, Proc","cited_arxiv_id":null,"evidence_quote":"supplies the distance associated with the metric $ds=\\sqrt G\\,dr$ used to interpret the exponent in Theorem 1.4"}],"review_version":1}