Counterexamples to the Landis conjecture in dimensions three and higher

Rupert L. Frank, Paata Ivanisvili

arXiv:2608.00802v1 Primary: math.AP Secondary: math-ph

Abstract

In every dimension three and higher we construct a nonzero function that satisfies the stationary Schrödinger 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.

Disclosure

“and FR 2664/3-1. P.I. acknowledges partial support from the US NSF CAREER grant DMS-2152401, US NSF grant DMS-2554183, a Simons Fellowship, and a Humboldt Research Fellowship for Experienced Researchers. The authors acknowledge the use of AI tools. All mathematical arguments and proofs in the final manuscript were checked and written by the authors. References [1] J. Bourgain and C. E. Kenig, On localization in the continuous Anderson–Bernoulli model in higher dimension, Inv”
PDF page 33

Classification: Rewriting existing author-written text
Multiplier: 4
Verified: 2026-08-23T19:16:44.064493Z

Structural counts

Pages 34 pdf

Theorems 1 source

Lemmas 27 source

Propositions 6 source

Corollaries 1 source

Definitions 0 source

Displayed equations 262 source

Bibliography entries 12 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file Landis_V3.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.