Sharp constructions of eigenfunctions of the magnetic Schr\"odinger operator

TL;DR

This paper establishes the sharpness of quantitative unique continuation estimates for solutions to a magnetic Schrödinger operator with decaying complex potentials, by constructing explicit examples that match the theoretical bounds.

Contribution

It constructs explicit solutions demonstrating the optimality of decay estimates for eigenfunctions of the magnetic Schrödinger operator with decaying potentials.

Findings

Constructed solutions match the decay bounds, proving sharpness.

Demonstrated the optimality of unique continuation estimates.

Extended results to complex-valued potentials and magnetic operators.

Abstract

We prove sharpness of quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + V u = \la u$, where $\la \in \C$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim <x>^{-N}$ and $|W(x)| \lesssim <x>^{-P}$. For $M(R) = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, it was shown in a companion paper that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \gtrsim \exp(-C R^{\be_0}(\log R)^{A(R)})$, where $\be_0 = max{2 - 2P, (4-2N)/3, 1}$. Under certain conditions on $N$, $P$, $\la$, and the dimension, we construct examples (some of which are in the style of Meshkov) to prove that this estimate for $M(R)$ is sharp. That is, we construct functions $u$, $V$ and $W$ such that $-\Delta u + W\cdot \nabla u + V u = \la u$, $|V(x)| \lesssim <x>^{-N},$ $|W(x)| \lesssim <x>^{-P}$ and $|u(x)| \lesssim \exp(-c|x|^{\be_0}(\log |x|)^C)$.