REVIEW 3 major objections 4 minor 10 references
Unique continuation at infinity for potentials with arbitrary radial growth
T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read 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 $\beta(r)=(n+1)r+\int_0^r G$.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [§2.4, Eq. (2.17)] 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.
- [§2.4, Eq. (2.19)] 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.
- [§2.4, Eqs. (2.22)–(2.25)] 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.
minor comments (4)
- [§2.3, Corollary 1.3] 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.
- [§2.4, Theorem 1.4] 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.
- [§2.4, after Eq. (2.21)] 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.
- [Various] 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.
Circularity Check
No circularity: the Landis-type bounds are derived from stated assumptions on G and V via a self-contained contradiction argument; the algebraic defects in Theorem 1.4 are correctness issues, not circularity.
full rationale
The paper's derivation chain does not reduce any claimed conclusion to its own inputs. In Theorem 1.1, the spherical mean w solves (2.4) with coefficients determined by n and V; the lower bound β' = n+1+G is fixed by the definition (1.1), and the proof only uses that definition plus |V| ≤ G to derive a contradiction from the assumed decay. No quantity is fitted to the solution u, and no prediction is renamed from a fitted parameter. Theorem 1.4 likewise proceeds by contradiction from the stated hypotheses G ∈ C^2, G ≥ c, |f''/f'|, |f'''/f'| bounded; the constants c1, c2, β are chosen from bounds on f and G, not from u or from the target lower bound. The cited works (Rossi, Davey, Meshkov, Logunov et al.) are background and are not load-bearing inputs; there are no self-citations. The printed proof of Theorem 1.4 does contain algebraic errors—equation (2.17) writes r^{-1} where f(r)^{-1} is required in the pullback of (2.16), and (2.23) omits the c2∫|v| term present in (2.22)—but those are repairable computational slips and do not constitute circular dependence of the theorem on itself. Because the central arguments are self-contained against external benchmarks and the assumptions do not include the conclusion, no circularity step is exhibited; the score is 0.
Assumptions & free parameters
assumptions (3)
- standard math Radial spherical average w(r) satisfies the second-order ODE w''=(1-n)r^{-1}w'+V(r)w and obeys unique continuation on [ϵ/2,∞).
- domain assumption V is radial and |V(x)|≤G(|x|) with G continuous; u is real-valued; u(0)≠0.
- domain assumption In Theorem 1.4, G∈C^2, G≥c>0, and for f=g^{-1} the ratios |f''/f'| and |f'''/f'| are bounded on [1,∞).
Cite this review
Pith. "Pith review of Unique continuation at infinity for potentials with arbitrary radial growth." pith.science (2026). https://pith.science/paper/T6NZRVQJ
@misc{pith2026260807276,
author = {Pith},
title = {Pith review of: Unique continuation at infinity for potentials with arbitrary radial growth},
year = {2026},
howpublished = {\url{https://pith.science/paper/T6NZRVQJ}},
note = {Machine review of arXiv:2608.07276}
}
abstract
Let $G$ be any given continuous positive function on $\mathbb{R}_+$. Let $V$ be radial with $|V(x)|\leq G(|x|)$. We prove a Landis-type theorem for any real-valued solution of $\Delta u=Vu$ on $\mathbb{R}^n$. We construct a decay threshold $e^{-g(r)}$, where $g$ is a strictly increasing function which can be computed explicitly in terms of $G$. Under suitable assumptions the exponent in the decay threshold is proportional to the Agmon distance associated with $G$.
Reference graph
Works this paper leans on
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[1]
Agmon, On exponential decay of solutions of second order ellitic questions in bounded domains, Proc
S. Agmon, On exponential decay of solutions of second order ellitic questions in bounded domains, Proc. A. Pleijel Conf., Uppsala, Sept. 1979
work page 1979
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[2]
On localization in the continuous Anderson-Bernoulli model in higher dimension
Bourgain, J., Kenig, C. On localization in the continuous Anderson-Bernoulli model in higher dimension . Invent. Math. 161 (2005), 389–426
work page 2005
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[3]
Davey, On Landis’ Conjecture in the Plane for Potentials with Growth , Viet
B. Davey, On Landis’ Conjecture in the Plane for Potentials with Growth , Viet. J. Math. 52 (2024), pp. 675–688
work page 2024
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[4]
Counterexamples to the Landis conjecture in dimensions three and higher
R. Frank, P. Ivanisvili, Counterexamples to the Landis conjecture in dimensions three and higher , arXiv:2608.00802
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[5]
E. M. Landis, Some questions in the qualitative theory of second-order elliptic equa- tions. Uspekhi Math. Nauk 8 (1953), no. 1 (49), pp. 8–31. 10 HENRIK UEBERSCH ¨AR
work page 1953
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[6]
E. M. Landis, Second-order equations of elliptic and parabolic type. Nauka, Moscow, 1971
work page 1971
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[7]
V. Z. Meshkov, On the possible rate of decay at infinity of solutions to second order partial differential equations. Math. USSR Sb. 72 (1992), 343–360
work page 1992
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Show all 10 references
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[9]
Logunov, E
A. Logunov, E. Malinnikova, N. Nadirashvilli, F. Nazarov, The Landis conjecture on exponential decay, Invent. Math. 241 (2025), pp. 465–508
2025
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[10]
Rossi, The Landis Conjecture with Sharp Rate of Decay , Indiana Univ
L. Rossi, The Landis Conjecture with Sharp Rate of Decay , Indiana Univ. Math. J. 70, No. 1 (2021), pp. 301–324. Sorbonne Universit ´e and Universit ´e Paris Cit ´e, CNRS, IMJ-PRG, F-75005 Paris, France. Email address : henrik.ueberschar@imj-prg.fr
2021
Reviewed August 10, 2026 · model on record in the stance chip above.
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