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

2024/01/05 by S. Krymskii, Krymskii, Stanislav · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2401.02800

openalex publication_date 2024/01/05 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28

Abstract

In this article we study the possible size of support of solutions to the discrete stationary Schrodinger equation Δu(x)+V(x)u(x)=0 in ℤ^d. We show that for any nonzero solution to any discrete stationary Schrodinger equation the dimension of the support is at least log₂(d)-7. In the related setting of ℤ₂-valued harmonic functions in ℤ^d one can improve the estimate on support's dimension to log₂(d). However, we also provide an example where a ℤ₂-valued harmonic function in ℤ^d has a fractal-like support with dimension log₂(d)+1. This fractal satisfies a recurrence relation: X = 2X+\e₁,-e₁,…,e_d,-e_d\. This example and estimate provide an answer to the Malinnikova's question about the smallest size of set X⊂ℤ^d such that no cross contains exactly one point of X.

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• Sparse supports of lattice eigenfunctions: quantitative growth and algebraic rigidity

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