Sharp bound on the largest positive eigenvalue for one-dimensional Schr\"odinger operators

Let $H=-D^2+V$ be a Schr\"odinger operator on $L^2(\mathbb{R})$, or on $L^2(0,\infty)$. Suppose the potential satisfies $\limsup_{x\to \infty}|xV(x)|=a<\infty$. We prove that $H$ admits no eigenvalue larger than $\frac{4a^2}{\pi^2}$. For any positive $a$ and $\lambda$ with $0<\lambda< \frac{4a^2}{\pi^2}$, we construct potentials $V$ such that $\limsup_{x\to \infty}|xV(x)|=a$ and the associated Sch\"rodinger operator $H=-D^2+V$ has eigenvalue $\lambda$.