An improved bound for the ground state of a Schr\"odinger operator on a loop

Abstract

Consider a closed curve of length $2\pi$ with curvature $\kappa(s)$ and the Schr\"odinger operator $H$ with $\kappa^2$ as the potential term. Let $\lambda_\Gamma$ be the lowest eigenvalue of $H$. The Ovals Conjecture proposed by Benguria and Loss states that $\lambda_\Gamma \ge 1$. While the conjecture remains open, the present work establishes a new lower bound of $0.81$ on $\lambda_\Gamma$, improving on the previously best known estimate of $0.6085$.