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On the lowest possible dimension of supports of solutions to the discrete Schrodinger equation

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arxiv 2401.02800 v1 pith:LW7PFZU4 submitted 2024-01-05 math.AP

Stanislav Krymskii This is my paper · ORCID
classification math.AP
keywords mathbbdimensionsupportdiscreteequationschrodingerestimateexample
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abstract

In this article we study the possible size of support of solutions to the discrete stationary Schrodinger equation $\Delta u(x)+V(x)u(x)=0$ in $\mathbb{Z}^d$. We show that for any nonzero solution to any discrete stationary Schrodinger equation the dimension of the support is at least $\log_2(d)-7.$ In the related setting of $\mathbb{Z}_2$-valued harmonic functions in $\mathbb{Z}^d$ one can improve the estimate on support's dimension to $\log_2(d).$ However, we also provide an example where a $\mathbb{Z}_2$-valued harmonic function in $\mathbb{Z}^d$ has a fractal-like support with dimension $\log_2(d)+1$. This fractal satisfies a recurrence relation: $$X = 2X+\{e_1,-e_1,\dots,e_d,-e_d\}.$$ This example and estimate provide an answer to the Malinnikova's question about the smallest size of set $X\subset\mathbb{Z}^d$ such that no cross contains exactly one point of $X$.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantitative Unique Continuation on Simplex and $\mathbb{Z}^n$

    math.AP 2026-08 conditional novelty 8.0 of 10

    For every n≥3, non-trivial solutions on the simplex lattice Δ_N^{(n)} and on Z^n have at least c N^{⌈n/2⌉} (resp. c L^{⌈n/2⌉}/log L) sites where the value is not exponentially small.

  2. Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity

    math.CA 2026-08 conditional novelty 8.0 of 10

    Harmonic functions on Z^d with u(0) nonzero have at least (10^(-10)/d) n^2 nonzero values in each n-cube, and this n^2 order is sharp in dimension three.

  3. Discrete Unique Continuation on Simplex

    math-ph 2026-08 accept novelty 6.0 of 10

    A nonzero balanced value in a function satisfying complete oriented-simplex cancellation forces support at least c_n R^{ceil(n/2)}, an exponent shown optimal by explicit examples.

  4. On Support Cardinality for the Discrete Schr\"odinger Equation

    math-ph 2026-06 unverdicted novelty 6.0 of 10

    Minimal support cardinality S_d(N) of nontrivial origin-nonzero Dirichlet solutions is nondecreasing in d, so S_4(N) ≳ N²/log N, matching even-dimensional constructions up to a log factor.

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