Some Schrödinger operators with dense point spectrum

Abstract:

Given any sequence $\{E_{n}\}^{\infty }_{n=1}$ of positive energies and any monotone function $g(r)$ on $(0,\infty )$ with $g(0)=1$, $\lim \limits _{r\to \infty } g(r)=\infty$, we can find a potential $V(x)$ on $(-\infty ,\infty )$ such that $\{E_{n}\}^{\infty }_{n=1}$ are eigenvalues of $-\frac {d^{2}}{dx^{2}}+V(x)$ and $|V(x)|\leq (|x|+1)^{-1}g(|x|)$.