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Eigenfunctions with double exponential rate of localization

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Pith's one-line read This paper constructs nonzero eigenfunctions of uniformly elliptic divergence-form operators that decay at double-exponential speed in a cylinder, the fastest rate allowed.

desk verdict 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. read the letter →

arxiv 2501.15354 v1 pith:LTVS4MPT submitted 2025-01-26 math.AP math.CAmath.SP

S.Krymskii , A.Logunov , F.Pagano This is my paper · ORCID
classification math.APmath.CAmath.SP MSC 35B6035J1535B4035P05
keywords Landisconjectureeigenfunctionlocalizationdoubleexponentialdecaydivergence-formellipticoperatorsuniquecontinuationcounterexamplesperiodicheatequationwithdriftAnderson
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 9.2, Remark 9.1 (Lemma 7.4)] 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.
  2. [Section 7.2.2, end of proof of Proposition 7.2] 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.
minor comments (4)
  1. [Section 5.3, Step 3] The text refers to 'By Lemma 6.3' where the statement is Proposition 6.3; please correct the cross-reference.
  2. [Section 9.2, Lemma 7.4] There is a typo in the lemma statement: 'sataisfies' should be 'satisfies'.
  3. [Section 7.3, Proposition 7.5] 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.
  4. [Section 1.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a self-contained iterative construction; the unproved Lemma 7.4 is a routine omitted verification, not a circular reduction.

full rationale

The derivation chain Theorem 1.2 -> Theorem 4.1 -> Theorem 5.1 -> Lemma 5.2 -> Lemma 5.3 -> Propositions 6.1-6.3 -> Proposition 7.2 -> Lemma 7.3 is a sequence of explicit reductions in which no parameter is fitted to the target decay and no result is imported from the authors' prior work as a load-bearing premise. The transfer constants c2, c3 and C are determined by the interface equalities (122), (140), (145) and by the explicit gluing functions; the double-exponential rate then follows from kn = 2^{n0+n-1}, tn = C(n-1), and the estimates (102)-(104). Lemma 7.3 is proved in Appendix 9.2 with an explicit vector field (343) satisfying div(V) = cos(k'y) by direct computation, giving the uniform bounds (340) that keep every glued matrix in the classes R(20,10) and R(80,60). Lemma 7.4, stated without proof in Remark 9.1, is the only missing support: the x/y-swap claimed there is not literal because k' differs from k, but the analogous vector field V = (2 sin(kx) sin^2(k'y)/k, sin(2k'y) cos(kx)/(2k')) has div(V) = cos(kx) and produces the same bounds, so the omitted proof is a routine verification rather than a circular reliance on the theorem being proved. The self-citation [14] (Filonov-Krymskii) appears only in the introduction as context on Landis-type counterexamples and is not used in any reduction. No fitted-input-as-prediction, self-definitional, or uniqueness-imported-from-authors pattern occurs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The construction rests on standard PDE machinery (uniform ellipticity, the maximum principle, classical barriers) plus an explicit family of smooth step functions theta; no data are fitted and no physical entities are introduced. The listed free parameters are hand-chosen universal constants (initial frequency scale, intermediate decay coefficient, perturbation size, acceleration window) that the final decay constants absorb; none is tuned to match a pre-specified target result. The only external inputs are the classical double-exponential barriers used for the optimality claims in Remark 1.4.

free parameters (4)
  • n0 (initial frequency exponent)
    k1 := 2^{n0} with n0 >> 1 chosen so that 2^{n0} >> sqrt(|mu|) (Section 5.1, step 3, eq. 75); the decay constants c(mu), C(mu) in Theorems 1.1 and 1.2 depend on it. A hand-chosen universal construction parameter, not fitted to data.
  • Intermediate decay coefficient b = 1/9 = 1/9
    In Lemma 5.3 Step 1, the coefficient is changed to diag(1, 1/9) so that the intermediate solution e^{-k't/3} decays slower than e^{-kt} (k'/3 < k for k' <= 2k). Any b with k' sqrt(b) < k and b in (1/10, 10) would work; 1/9 is a hand choice of convenience.
  • Perturbation size epsilon = 1/k^4 = 1/k^4
    Chosen in the reduction of Proposition 7.1 to Proposition 7.2 to make the wait-time t0 = 8 ln k/(k sqrt(a) - k' sqrt(b)) work with epsilon^{-1/4} <= k, k' <= epsilon^{-1/3}; other sufficiently small powers of 1/k would serve.
  • Acceleration window w = 400 = 400
    Universal constant chosen in Proposition 6.3 to bound |d(b-tilde)/dt| <= 10; any sufficiently large universal w works. Chosen in the proof below eq. 253.
assumptions (4)
  • standard math Uniform ellipticity of the 3D operator u_tt + div(A grad u) together with the maximum principle controls the sup-norm of glued solutions.
    Invoked in Section 9.1, eq. 337, to obtain sup_{[0,2w]} |u| = 1 for the transformed solution in Lemma 2.1; the 3x3 matrix diag(A,1) is uniformly elliptic because A is.
  • standard math Landis barrier: elliptic solutions cannot decay strictly faster than double-exponentially, and the parabolic analogue holds.
    Invoked in Remark 1.4 to claim optimality of the constructed decay, citing [25] for the elliptic case and [5] for the parabolic case; the exact coefficient-class match with those theorems is not spelled out (flagged).
  • standard math Smooth cutoff functions theta with flat endpoints exist and satisfy the derivative bounds used throughout (|theta'| <= sqrt(pi), sup |theta^{(n)}| <= C_n).
    The concrete theta is defined in Section 2 (footnote) via the Gaussian error function and is used in Propositions 6.1, 6.2, 6.3, 7.2 and Lemma 8.2; it is the standard smooth step function with all derivatives vanishing at 0 and 1.
  • domain assumption Unique continuation holds for C1-coefficient elliptic operators, so the constructed fast-decaying solutions must not vanish on any open set.
    Section 1.3 frames the coefficient-regularity landscape via Garofalo-Lin unique continuation; the construction respects this constraint by design, producing double-exponential decay rather than compact support for C1 coefficients.

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Pith. "Pith review of Eigenfunctions with double exponential rate of localization." pith.science (2026). https://pith.science/paper/LTVS4MPT

@misc{pith2026250115354,
  author       = {Pith},
  title        = {Pith review of: Eigenfunctions with double exponential rate of localization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LTVS4MPT}},
  note         = {Machine review of arXiv:2501.15354}
}
abstract

We construct a real-valued solution to the eigenvalue problem $-\text{div}(A\nabla u)=\lambda u$, $\lambda>0,$ in the cylinder $\mathbb{T}^2\times \mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that $|u(x,y,t)|\leq C e^{-c e^{c|t|}}$ for some $c,C>0$. We also construct a complex-valued solution to the heat equation $u_t=\Delta u + B \nabla u$ in a half-cylinder with continuous and uniformly bounded $B$, which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Miller, are reviewed.

Figures

Figures reproduced from arXiv: 2501.15354 by the authors.

Figure 1
Figure 1. Transforming f into g 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. A graph of the function θ Lemma 2.1. Consider two harmonic functions in T 2 × R f := cos(kx)e −kt and g = cos(k ′ y)e −k ′ t and assume that the positive numbers k, k′ , w > 0 satisfy 0 < k′ − k ≲ w −1 ≲ k. (12) Put α(t) := θ(t/w) and u := ( f + [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Main step 31 [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Changing a coefficient Step 2: transformation into a slower-decaying solution Define u˜2 := cos(k ′ y)e −k ′ 3 t (138) and note that it is a solution to u¨ + div 1 0 0 1 9  ∇u  = 0 and that u˜2 oscillates faster but decays slower that u1 = cos(kx)e −kt defined in t…
Figure 5
Figure 5. Figure 5: Transformation into a slower decaying but faster oscillating solution [PITH_FULL_IMAGE:figures/full_fig_p034_5.png]
Figure 6
Figure 6. Figure 6: Acceleration Conclusion • We define the total duration C2 := 2 + C1 ≥ 2 which is independent of k, k′ (as claimed in Lemma 5.3). We have been able to transform cos(kx)e −kt into c3 cos(k ′y)e −k ′ t via a solution u to u¨ + div(A∇u) = 0. We saw that A is uniformly elli…

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