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

TL;DR

This paper establishes a sharp upper bound on the largest eigenvalue of one-dimensional Schrödinger operators with potentials satisfying a specific growth condition, and constructs potentials that attain eigenvalues below this bound.

Contribution

It provides the first sharp bound on the maximum eigenvalue for such Schrödinger operators and demonstrates the attainability of eigenvalues below this bound through explicit potential construction.

Findings

No eigenvalue exceeds 4a^2/π^2 for potentials with limsup |xV(x)|=a.

Existence of potentials with eigenvalues arbitrarily close to the bound.

Explicit construction of potentials achieving eigenvalues below the bound.

Abstract

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 $\lambda$.