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

Stanislav Krymskii

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 $\mathbb{Z}^d$. 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 $X$.

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

Abstract

In this article we study the possible size of support of solutions to the discrete stationary Schrodinger equation in . We show that for any nonzero solution to any discrete stationary Schrodinger equation the dimension of the support is at least In the related setting of -valued harmonic functions in one can improve the estimate on support's dimension to However, we also provide an example where a -valued harmonic function in has a fractal-like support with dimension . This fractal satisfies a recurrence relation: This example and estimate provide an answer to the Malinnikova's question about the smallest size of set such that no cross contains exactly one point of .
Paper Structure (7 sections, 13 theorems, 106 equations)

This paper contains 7 sections, 13 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Example
  3. Optimality in the harmonic case
  4. Optimality for the Malinnikova question
  5. Start of proof
  6. The main lemma in the proof of Theorem \ref{['mainmal']}
  7. End of proof of Theorem \ref{['mainmal']}

Key Result

Theorem 2.1

There exists a $\mathbb{Z}_2$-valued harmonic function $u$ defined on the set $\mathbb{Z}^d$ such that there is a constant $C = C(d)$ s.t. for any $n\in\mathbb{N}$ there is no ball of size $n$ containing more than $Cn^{\log_2(2d)}$ points of $\operatorname{supp} u.$

Theorems & Definitions (31)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • proof : Proof of Theorem \ref{['2.1']}
  • ...and 21 more