Barry Simon’s L² conjecture (Problem 8 of his 2000 list) disproved
Becker and Yang build a counterexample largely found with ChatGPT
Awaiting review
Importance: major (4 of 5)The takeaway
One of Barry Simon’s fifteen problems for Schrödinger operators, open since 2000, is false: a small, smooth potential satisfying his L² condition can leave no continuous spectrum at all. The authors say ChatGPT produced the key mechanism and, ‘very much to our surprise’, extended it to the full spectrum.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Expert-checked
- Importance
- Major (4 of 5)
- Last verified
- 23 October 2026
Your AI and this story
- GPT-6 Astra175 days after its cutoff
- Claude Opus 5.5114 days after its cutoff
- Gemini 4 Argon205 days after its cutoff
- Grok 4.7144 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 114 days before it.
Key facts
- Conjecture (Simon 2000, Problem 8; Simon 2019, Conjecture 20.2): if ∫|x|^{1−d}|V(x)|² dx < ∞ then −Δ + V has absolutely continuous spectrum of infinite multiplicity at almost every positive energy. The 1D analogue is the Deift–Killip theorem (1999)
- Theorem 1.1: for d ≥ 2, 0 ≤ σ < 1/3 and δ > 0 there is a smooth bounded nonnegative V with ‖V‖∞ + I_{d,0}(V) + I_{d,σ}(V) < δ such that σ(H_V) = σ_ess(H_V) = [0, ∞) and the continuous spectral projection is zero; the positive eigenvalues are dense
- Construction: potentials on widely separated annuli, each a weak angular ‘lens’ that focuses waves into narrow channels plus bounded periodic ‘reflecting barriers’ whose spectral gaps cover a prescribed energy window; pure point spectrum via Combes–Hislop spectral averaging; 3D by cylindrical extension, all d by tensor sums
- AI disclosure (verbatim): ‘Our interest in this conjecture was renewed when it remained among the few of Simon’s problems left untouched by OpenAI … We developed the first construction of the potential in discussions with ChatGPT, obtaining a proof of absence of absolutely continuous spectrum in a bounded interval … When prompted with this question, ChatGPT showed that a rather intricate adjustment of the parameters of the potential … could yield point spectrum in a bounded interval through a mechanism very similar to the one used in the final proof and already including the spectral-averaging argument. Very much to our surprise, ChatGPT was then able to extend this result to the full spectrum by pushing the parameters of the construction further. We thoroughly rewrote the resulting argument and manuscript’
- ChatGPT model version not stated. Simon Becker earlier used GPT-6.1 Sol for counterexamples to Fried’s conjecture (October 2026)
What happened
In 2000 Barry Simon published fifteen open problems for Schrödinger operators. Problem 8 asked whether the condition ∫|x|^{1−d}|V|² < ∞, which by the Deift–Killip theorem guarantees absolutely continuous spectrum in one dimension, does the same in higher dimensions. Simon restated it as Conjecture 20.2 in 2019.
Simon Becker and Mengxuan Yang (arXiv 2610.316371, 22 Oct) show it is false in every dimension d ≥ 2. Their potential is a sum of annular blocks: a weak “lens” bends waves into narrow channels that end at bounded periodic barriers whose spectral gaps cover a given energy window. Spectral averaging then gives pure point spectrum, dense in [0, ∞), so the operator has no continuous spectrum at all, even under the stronger weighted condition with |x|^σ, σ < 1/3.
The authors write that they returned to the problem because it “remained among the few of Simon’s problems left untouched by OpenAI”. ChatGPT produced the first construction, then the mechanism used in the final proof, and finally the extension to all energies. The authors rewrote the argument and take full responsibility for it.
No expert reactions were found on 23 Oct (X and blogs not searched).
Why it matters
Simon’s 2000 list is one of the standard problem lists of mathematical physics. This is a clean negative answer to one of its problems, with the AI credited for the decisive steps, not just for checking. It also shows researchers choosing targets by what OpenAI’s 6 Oct release did not cover.
What is disputed or not yet verified
| Verification | Unrefereed arXiv preprint (2610.31637); checked by its authors |
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Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed (PDF read)