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

Abstract

For every $d\ge 2$ and $0\le \sigma <1/3$, we construct with LLM-assistance a smooth bounded nonnegative lens-type potential $V$ with arbitrarily small $\Vert V{\Vert }_{\infty }+{\int }_{{\mathbb{R}}^d}|x{|}^{1-d+\sigma }|V(x){|}^{2}dx$, such that $-\Delta +V$ 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.

arXiv comment35 pages, 1 figure. Comments are welcome