An improved bound for the ground state of a Schrödinger operator on a loop

Abstract

Consider a closed curve of length $2\pi$ with curvature $\kappa (s)$ and the Schrödinger 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 $0.6085$.