For radial potentials with arbitrary growth, any real-valued solution of Δu=Vu with u(0)≠0 has a sequence of points where |u(x)|≥c e^{-β(|x|)}, with β explicit in terms of the growth bound.
Counterexamples to the Landis conjecture in dimensions three and higher
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In every dimension three and higher we construct a nonzero function that satisfies the stationary Schr\"odinger equation with bounded, real-valued potential and decays like $\exp(-c|x|^{4/3})$. This gives a counterexample to the Landis conjecture in dimension three and higher.
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Unique continuation at infinity for potentials with arbitrary radial growth
For radial potentials with arbitrary growth, any real-valued solution of Δu=Vu with u(0)≠0 has a sequence of points where |u(x)|≥c e^{-β(|x|)}, with β explicit in terms of the growth bound.