Breakthrough Tracker record
An AI-assisted preprint claims a counterexample to Simon's multidimensional L2 conjecture
Becker and Yang construct, for every dimension at least two and every weight exponent below one third, smooth bounded nonnegative potentials with arbitrarily small weighted L2 size whose Schrödinger operators have pure point spectrum throughout the positive axis. If correct, the construction disproves Barry Simon's Problem 8.
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- Stable ID
math-simon-l2-conjecture-counterexample-2026- Revision
math-simon-l2-conjecture-counterexample-2026.v1- Field
- Mathematics · Spectral theory and Schrödinger operators
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed counterexample
- Last checked
AI role
The authors say the first construction emerged in discussions with ChatGPT, which later supplied the parameter adjustment leading to point spectrum and the extension to the full positive spectrum. They rewrote the argument and take responsibility for the manuscript; the model version is not disclosed.
Record details
- Problem or result
- Barry Simon's multidimensional weighted-L2 conjecture for preservation of absolutely continuous spectrum
- Authors
- Simon Becker and Mengxuan Yang
- Institutions
- Bocconi University; Texas A&M University
- Result date
- First-version preprint submitted October 22, 2026
Why it matters
The conjecture sought a broad integrability condition guaranteeing that multidimensional Schrödinger operators retain continuous spectrum. A valid counterexample would rule out that program even under a stronger positive spatial weight and with arbitrarily small bounded potentials.
Limits
This is a new unrefereed preprint with no independent specialist correctness assessment located in this cycle. The construction is highly parameter-sensitive, and the tracker has not reproduced its spectral-averaging argument. The record therefore says claimed counterexample, not proved theorem.
Sources
Correction and revision history
- 2026-10-25 — Added after primary-source, scope, status, AI-role and limitation review.
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