Harmonic rigidity at fixed spectral gap in one dimension

TL;DR

This paper identifies the harmonic trap as the unique potential maximizing ground-state position variance among one-dimensional confining potentials with a fixed spectral gap, establishing a sharp quantum speed-limit and stability results.

Contribution

It provides the first rigorous proof of harmonic rigidity at fixed spectral gap, including quantitative stability and extensions to magnetic settings.

Findings

Harmonic trap uniquely maximizes ground-state position variance.

Established a sharp quantum speed-limit bound on the quantum metric.

Extended analysis to magnetic potentials and applications to polarizability and potential benchmarking.

Abstract

We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.