Some Quantitative Unique Continuation Results for Eigenfunctions of the Magnetic Schrödinger Operator

We prove quantitative unique continuation results for solutions of -Δu + W · ∇u + Vu = λu, where λ ∈ ℂ and V and W are complex-valued decaying potentials that satisfy {pipe}V(x){pipe} < 〈x〉^-N and {pipe}W(x){pipe} < 〈x〉^-P. For (Formula Presented), we show that if the solution u is non-zero, bounded, and u(0) = 1, then (Formula presented), where. Under certain conditions on N, P and λ, 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 (Formula presented).