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 $\Delta u(x)+V(x)u(x)=0$ 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 $X$.