Breakthrough Tracker record

A preprint gives real-valued counterexamples to Landis’s conjecture in dimensions three and higher

Frank and Ivanisvili construct, in every dimension at least three, a nonzero solution of the stationary Schrödinger equation with bounded real-valued potential that decays like exp(−c|x|^(4/3)), giving the claimed negative answer to Landis’s conjecture in those dimensions.

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Stable ID: math-landis-conjecture-real-counterexamples-2026
Revision: math-landis-conjecture-real-counterexamples-2026.v1
Field: Mathematics · Analysis of partial differential equations and mathematical physics
Evidence: Tier 1 · Peer reviewed: No
Record state: Provisional · Provisional counterexamples
Last checked: 2026-08-08

AI role

The authors acknowledge using AI tools but state that they checked and wrote every mathematical argument and proof in the final manuscript. The disclosure does not specify the tools or individual contributions.

Record details

Problem or result: Landis’s unique-continuation conjecture for real-valued Schrödinger potentials
Authors: Rupert L. Frank and Paata Ivanisvili
Institutions: LMU Munich and Munich Center for Quantum Science and Technology; University of California, Irvine
Result date: Preprint submitted August 1, 2026

Why it matters

The exponent four-thirds is the sharp complex-valued decay scale from Meshkov’s construction. Producing the same scale with a real potential resolves the higher-dimensional form of a long-standing unique-continuation question.

Limits

This is a 34-page first-version preprint without located independent specialist assessment or peer review. The construction applies in dimensions three and higher; the real-valued two-dimensional problem remains open.

Sources

Primary: Frank and Ivanisvili, Counterexamples to the Landis conjecture in dimensions three and higher

Correction and revision history

2026-08-08 — Added after primary-source review as a provisional claim; the scope, evidence level and material limitations are stated explicitly.

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