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

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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 |V(x)| x⁻ⁿ and |W(x)| x⁻ᵖ. For M(R) = |ₓ₀| ₌ ᵣ||u||ₗ₂₍B₁₍ₓ₀₎₎, we show that if the solution u is non-zero, bounded, and u(0) = 1, then M(R) (-C Rβ⁰(R)ᵃ⁽ ʳ⁾), where β₀ = {2 - 2P, 4-2N/3, 1}. 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 -Δu + W· ∇ u + Vu = λu, |V(x)| x⁻ⁿ, |W(x)| x⁻ᵖ and |u(x)| (-c|x|β⁰(|x|)ᶜ).