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

For every $d\ge 2$ and $0\le \sigma <1\mathrm{/}3$, we construct with LLM-assistance a smooth bounded nonnegative lens-type potential $V$ with arbitrarily small $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}+{\int }_{{\mathbb{R}}^d}\mathrm{\mid }x{\mathrm{\mid }}^{1-d+\sigma }\mathrm{\mid }V(x){\mathrm{\mid }}^{2}dx$, such that $-\mathrm{\Delta }+V$ has only pure point spectrum in $(0,\mathrm{\infty })$. Thus, we show that Barry Simon’s $L^2$ conjecture, Problem 8 in “Schrödinger operators in the twenty-first century”, is false.