AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-23
Lenses make many points: Pure point spectrum disproves Simon's $L^2$ conjecture
With LLM assistance, the authors construct, for every dimension d≥2 and 0≤σ<1/3, a smooth bounded nonnegative lens-type potential whose norm can be made arbitrarily small. For this potential, the operator −Δ+V has only pure point spectrum in (0,∞). This shows that Barry Simon's L² conjecture, Problem 8 in Schrödinger Operators in the Twenty-First Century, is false.