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arXiv · 2610.31637

Lenses make many points: Pure point spectrum disproves Simon's $L^2$ conjecture

Simon Becker·Mengxuan Yang
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Abstract

For every $d\ge2$ and $0\leσ<1/3$, we construct with LLM-assistance a smooth bounded nonnegative lens-type potential $V$ with arbitrarily small $\|V\|_\infty+\int_{\mathbb{R}^d}|x|^{1-d+σ}|V(x)|^2 dx$, such that $-Δ+V$ has only pure point spectrum in $(0,\infty)$. Thus, we show that Barry Simon's $L^2$ conjecture, Problem 8 in "Schrödinger operators in the twenty-first century", is false.

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Simon Becker, Mengxuan Yang. 2026-10-22. Lenses make many points: Pure point spectrum disproves Simon's $L^2$ conjecture. https://arxiv.org/abs/2610.31637

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