On Support Cardinality for Discrete Schr\"odinger Equation

How sparse can a nontrivial solution of a discrete Schrödinger 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.