Sharp bound on the largest positive eigenvalue for one-dimensional Schrödinger operators

Abstract

Let $H=-D^2+V$ be a Schrödinger operator on $L^2(\mathbb{R})$, or on $L^2(0,\infty )$. Suppose the potential satisfies ${\mathrm{limsup}}_{x\to \infty }|xV(x)|=a<\infty$. 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 ${\mathrm{limsup}}_{x\to \infty }|xV(x)|=a$ and the associated Schrodinger operator $H=-D^2+V$ has eigenvalue $\lambda$.

arXiv commentAfter we finished this note, we noticed that the main result has been proved by Halvorsen and Atkinson-Everitt. So this paper is not intended for publication

References in corpus (1)

• Tosio Kato's Work on Non--Relativistic Quantum Mechanics

Cited by in corpus (1)

• Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators