A further quantification of the unique continuation properties of eigenfunctions of the magnetic Schrödinger operator

Abstract

We prove quantitative unique continuation results for solutions of $\Delta w-k^{2}w=Vw+W\cdot \nabla w$ in a neighborhood of infinity, where $k>0$, and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)|\lesssim |x{|}^{-N}$ and $|W(x)|\lesssim |x{|}^{-P}$ for some $N,P>1$. For $M(R,4n/k)=\inf \{||w|{|}_{L^2(B_{4n/k}(x_0))}:|x_0|=R\}$, we show that if the solution $w$ is non-zero, bounded, and normalized, then $M(R,4n/k)\gtrsim \exp (-kR-G\log R)$, where $G>\frac{n-1}{2}$ is a constant. An examination of radial solutions to $\Delta w-k^{2}w=Vw+W\cdot \nabla w$ shows that this new estimate for $M(R,4n/k)$ is sharp up to logarithmic terms.

This paper has been withdrawn due to an error in the set up of the problem

References in corpus (1)

• Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator