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

Abstract

We prove quantitative unique continuation results for solutions of $-\Delta u+W\cdot \nabla u+Vu=\lambda u$, where $\lambda \in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)|\lesssim ⟨x{⟩}^{-N}$ and $|W(x)|\lesssim ⟨x{⟩}^{-P}$. For $M(R)={\inf }_{|x_0|=R}||u|{|}_{L^2(B_1(x_0))}$, we show that if the solution $u$ is non-zero, bounded, and $u(0)=1$, then $M(R)\gtrsim \exp (-C{R}^{{\beta }_0}(\log R{)}^{A(R)})$, where ${\beta }_0=\max \{2-2P,\frac{4-2N}{3},1\}$. Under certain conditions on $N$, $P$ and $\lambda$, 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 $-\Delta u+W\cdot \nabla u+Vu=\lambda u$, $|V(x)|\lesssim ⟨x{⟩}^{-N}$, $|W(x)|\lesssim ⟨x{⟩}^{-P}$ and $|u(x)|\lesssim \exp (-c|x{|}^{{\beta }_0}(\log |x|{)}^C)$.

Final version as it appears in Communications in Partial Differential Equations

References in corpus (2)

• Unique Continuation for Schrödinger Evolutions, with applications to profiles of concentration and traveling waves
• Quantitative uniqueness for Schrodinger operator with regular potentials

Cited by in corpus (17)

• Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators
• Quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms
• Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian
• On Landis' conjecture in the plane when the potential has an exponentially decaying negative part
• The Landis Conjecture for variable coefficient second-order elliptic PDES
• On Landis Conjecture for the Fractional Schrödinger Equation
• On Landis' conjecture in the plane
• Quantitative uniqueness estimates for $p$-Laplace type equations in the plane
• Quantitative uniqueness estimates for second order elliptic equations with unbounded drift
• Landis-type conjecture for the half-Laplacian
• On the Fractional Landis Conjecture
• A further quantification of the unique continuation properties of eigenfunctions of the magnetic Schrödinger operator
• Quantitative uniqueness of some higher order elliptic equations
• Quantitative uniqueness of solutions to parabolic equations
• Asymptotic behavior of the steady Navier-Stokes flow in the exterior domain
• Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients
• Quantitative uniqueness estimates for the general second order elliptic equations