Counterexamples to the Landis conjecture in dimensions three and higher

Paata Ivanisvili; Rupert L. Frank

Counterexamples to the Landis conjecture in dimensions three and higher

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arxiv 2608.00802 v1 pith:OADUGUCD submitted 2026-08-01 math.AP math-phmath.MP

Rupert L. Frank , Paata Ivanisvili This is my paper · ORCID

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keywords higherthreeconjecturedimensionlandisboundedconstructcounterexample

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abstract

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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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.

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