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arXiv · 2504.20229

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

Helmut Linde
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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 $0.6085$.

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Helmut Linde. 2025-04-28. An improved bound for the ground state of a Schr\"odinger operator on a loop. https://arxiv.org/abs/2504.20229

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