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Minimal support size of discrete Schrödinger solutions cannot decrease when dimension rises, so lower bounds carry upward and give a nearly sharp four-dimensional estimate.

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2026-07-14 17:24 UTC pith:D7MEXB2K

load-bearing objection Clean, elementary dimension-reduction for support size of discrete Schrödinger solutions; upgrades the known 3-d lower bound to 4-d and is worth a short note.

arxiv 2606.26149 v2 pith:D7MEXB2K submitted 2026-06-23 math-ph math.MPmath.PR

On Support Cardinality for the Discrete Schr\"odinger Equation

Linjun Li This is my paper
classification math-ph math.MPmath.PR MSC 35J1039A1281Q10
keywords discrete Schrödinger equationsupport cardinalitydimension reductionunique continuationlattice LaplacianAnderson localizationDirichlet solutions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks how sparse a nontrivial solution of a discrete Schrödinger equation on a finite lattice box can be when an arbitrary real potential is allowed. Sparsity is measured by the number of sites where the solution is nonzero, under the requirement that it is nonzero at the origin and vanishes outside the box. The central claim is a dimension-reduction principle: that minimal support size is non-decreasing in the ambient dimension. Any lower bound already proved in dimension d−1 therefore transfers automatically to every higher dimension. Combining this transfer with an earlier three-dimensional lower bound immediately yields a nearly sharp lower bound in four dimensions that matches the best explicit constructions up to a logarithmic factor. The result matters because quantitative control on support size is a discrete substitute for unique continuation and is a known technical ingredient in proofs of Anderson localization for Bernoulli potentials.

Core claim

For every dimension d≥2 and every box size N, the minimal possible support cardinality S_d(N) of a real Dirichlet solution of −Δu+Vu=0 that is nonzero at the origin satisfies S_d(N)≥S_{d−1}(N). Consequently every lower bound known in dimension d−1 becomes a lower bound in dimension d, and in particular the three-dimensional bound of order N^{2}/log N lifts to four dimensions.

What carries the argument

Fiber-projection map F(u): each solution u on the d-dimensional box is sliced into one-dimensional fibers; a single fixed vector a is chosen so that the scalar products of a against those fibers produce a function F(u) that itself solves a discrete Schrödinger equation on the (d−1)-dimensional box and whose support is no larger than that of u.

Load-bearing premise

A single fixed vector can simultaneously detect every nonzero one-dimensional fiber of an arbitrary solution, because a finite union of hyperplanes cannot cover the whole Euclidean space.

What would settle it

Exhibit an explicit real solution on a four-dimensional box whose support has cardinality o(N^{2}/log N) while remaining nonzero at the origin; any such example would falsify the transferred lower bound.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any future improvement of the three-dimensional support lower bound automatically improves the four-dimensional bound by the same factor.
  • The same reduction shows that the conjectured optimal exponent ⌈d/2⌉ is already optimal once it is verified in the two lowest dimensions.
  • Support-cardinality lower bounds of this type continue to serve as quantitative unique-continuation substitutes needed for Bernoulli-Anderson localization arguments.
  • Explicit constructions of size roughly N^{d/2} remain admissible upper bounds and are now known to be nearly optimal through dimension four.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The fiber-projection idea may extend beyond the lattice Laplacian to other nearest-neighbor graph operators whose fibers remain one-dimensional.
  • If a random Bernoulli potential forces the support to become fully d-dimensional with high probability, the deterministic reduction still supplies the deterministic baseline against which the random improvement is measured.
  • The same monotonicity may apply to continuous analogues once a suitable notion of fiber projection is formulated, though the paper itself stays discrete.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies the minimal support cardinality S_d(N) of a nontrivial real solution u of the discrete Schrödinger equation −Δu + V u = 0 on the finite box B_d(N) with Dirichlet boundary conditions and u(0) ≠ 0, for an arbitrary real potential V. The main result (Theorem 1.1) is the monotonicity S_d(N) ≥ S_{d−1}(N) for d ≥ 2. The proof proceeds by viewing B_d(N) as a product of B_{d−1}(N) with a one-dimensional fiber, selecting a single vector a ∈ R^{2N+1} that detects every nonzero fiber (Lemma 2.1), and applying the linear functional fiberwise to obtain a (d−1)-dimensional solution F(u) whose support is no larger than that of u (Lemma 2.2). Combined with the known three-dimensional lower bound of Li–Zhang, this yields S_4(N) ≳ N^{2}/log N, which is nearly sharp against the explicit constructions of Proposition 2.3 that give upper bounds of order (2N+1)^{⌈d/2⌉}. A conjecture that the same exponent is optimal in all dimensions is stated.

Significance. The dimension-reduction principle is elementary, self-contained, and immediately upgrades every lower bound from dimension d−1 to all higher dimensions. In particular it produces the first nearly sharp support lower bound in four dimensions, matching the best available two-dimensional constructions up to a single logarithmic factor. The argument relies only on finite-dimensional linear algebra and the definition of the discrete Laplacian; no continuum limit or external analytic machinery is required beyond the already-published three-dimensional estimate used as a black box. The note therefore supplies a clean, reusable tool for quantitative unique-continuation questions on the lattice and for related Anderson–Bernoulli localization arguments.

minor comments (4)
  1. Lemma 2.1 is stated for x ∈ B_d(N−1); the subsequent text and the definition of F(u) make clear that the correct domain is B_{d−1}(N). The slip is purely notational and does not affect the argument.
  2. In the paragraph immediately after Lemma 2.2 the set B_{d−1}(B) appears twice; it should be B_{d−1}(N).
  3. The constant potential V ≡ −2d used for even dimensions in Proposition 2.3 is written both as −2d and as −4k; a single consistent expression would improve readability.
  4. A brief remark that the same vector a works simultaneously for every fiber of a fixed u (because the set of fibers is finite) would make the appeal to the finite-union-of-hyperplanes argument completely explicit for non-specialist readers.

Circularity Check

0 steps flagged

Main dimension-reduction theorem is elementary and self-contained; only a non-load-bearing self-citation appears in the 4-d numerical application.

full rationale

Theorem 1.1 (S_d(N) ≥ S_{d-1}(N)) is proved entirely inside Section 2 from first principles. Lemma 2.1 selects a single vector a that separates nonzero fibers by the elementary fact that a finite union of proper hyperplanes cannot cover R^{2N+1}; Lemma 2.2 then applies the linear functional fiberwise to the Schrödinger equation, defines the quotient potential V_1 where the residual vanishes whenever F(u)=0, and obtains a (d-1)-dimensional solution whose support is at most as large as that of u. No parameters are fitted, no uniqueness theorem is imported, and no external black-box enters the reduction itself. The only self-citation is the 3-d lower bound S_3(N) ≥ c N²/log N taken from the author’s earlier joint work [LZ22] and used solely to instantiate the numerical corollary S_4(N) ≥ c N²/log N; that citation is not required for the proof of the monotonicity principle and does not force the main claim. Upper-bound constructions (Proposition 2.3) are likewise explicit and independent. Consequently the derivation chain contains no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper is pure mathematics. It rests only on the standard definition of the discrete Laplacian with Dirichlet boundary conditions, elementary linear algebra (finite unions of hyperplanes do not cover R^m), and one external quantitative unique-continuation theorem used solely for the four-dimensional corollary. No free parameters or new physical entities are introduced.

axioms (3)
  • standard math A finite union of proper linear hyperplanes in R^m does not cover R^m, so a separating vector a exists for any finite collection of nonzero fibers.
    Invoked without proof in the argument of Lemma 2.1; classical fact from linear algebra.
  • domain assumption The discrete Laplacian on the box B_d(N) is defined by nearest-neighbor differences with zero exterior values.
    Standard setup of the discrete Schrödinger equation stated in the introduction and used throughout Section 2.
  • domain assumption S_3(N)≥c N²/log N for an absolute c>0 (Li–Zhang, Theorem 5.1).
    Imported only for the four-dimensional corollary; not used in the proof of the main reduction theorem.

pith-pipeline@v1.1.0-grok45 · 13333 in / 2099 out tokens · 35729 ms · 2026-07-14T17:24:59.018220+00:00 · methodology

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read the original abstract

How sparse can a nontrivial solution of a discrete Schr\"odinger equation be? In this note we study Dirichlet solutions on a finite $d$-dimensional lattice box, allowing an arbitrary real potential, and measure sparsity by the number of lattice sites at which the solution is nonzero (assuming it is nonzero at the origin). Our main result is a dimension-reduction principle: the minimal possible support size cannot decrease when the dimension increases. Consequently, any lower bound proved in dimension $d-1$ automatically yields the same lower bound in dimension $d$. As an application, we obtain a nearly sharp lower bound in four dimensions, matching the best-known two-dimensional constructions up to a logarithmic factor.

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Works this paper leans on

5 extracted references · 5 linked inside Pith

  1. [1]

    Buhovsky, A

    [BLMS17] L. Buhovsky, A. Logunov, E. Malinnikova, and M. Sodin. A discrete harmonic function bounded on a large portion ofZ 2 is constant. 2017, arXiv:1712.07902. [BYY20] P. Bourgade, H.-T. Yau, and J. Yin. Random band matrices in the delocalized phase I: quantum unique ergodicity and universality. Commun. Pure Appl. Math., 73(7):1526–1596,

    Pith/arXiv arXiv 2017

  2. [2]

    Elboim, A

    [EGH25] D. Elboim, A. Gloria, and J. Hernandez. The mirror walk in high dimensions. 2025, arXiv:2504.10615. [EKYY13] L. Erdős, A. Knowles, H.-T. Yau, and J. Yin. Delocalization and diffusion profile for random band matrices.Commun. Math. Phys., 323(1):367–416,

    Pith/arXiv arXiv 2025

  3. [3]

    Fernández-Bertolin, D

    [FBSR24] A. Fernández-Bertolin, D. Stan, and L. Roncal. Landis’ conjecture: a survey.arXiv preprint, page arXiv:2412.00788,

    Pith/arXiv arXiv

  4. [4]

    Krymskii

    [Kry24] S. Krymskii. On the lowest possible dimension of supports of solu- tions to the discrete schrodinger equation. 2024, arXiv:2401.02800. [Li20] L. Li. Polynomial bound for the localization length of lorentz mirror model on the 1d cylinder. 2020, arXiv:2010.05900. [Li21] L. Li. On the Manhattan pinball problem.Electronic Communica- tions in Probabili...

    Pith/arXiv arXiv 2024

  5. [5]

    Yau and J

    [YY25] H.-T. Yau and J. Yin. Delocalization of one-dimensional random band matrices. 2025, arXiv:2501.01718. [YYY22] F. Yang, H.-T. Yau, and J. Yin. Delocalization and quantum diffu- sion of random band matrices in high dimensions II: T-expansion. Commun. Math. Phys., 396(2):527–622,

    Pith/arXiv arXiv 2025