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

By Blair Davey. 2012

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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) = inf|x 0| = R ‖u‖ L 2(B 1(x 0)), we show that if the solution u is non-zero, bounded, and u(0) = 1, then M(R) ≳ exp(−CR β0 (log R) A(R)), 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 −Δu + W · ∇u + Vu = λu, |V(x)| < ⟨x⟩−N , |W(x)| < ⟨x⟩−P and |u(x)|

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3cc8d7c9615e98f9679be192defd96c391eb8284