Center-of-Mass Bounds and Harmonic Extremality

• URL: http://arxiv.org/abs/2607.08096v1
• Date: Thu, 09 Jul 2026 04:07:10 GMT
• Title: Center-of-Mass Bounds and Harmonic Extremality
• Abstract summary: We study the center-of-mass observable in one-dimensional many-body systems with translation-invariant interactions.<n>We extend the harmonic-rigidity mechanism from the one-body setting to an interacting many-body problem.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the center-of-mass observable in one-dimensional many-body systems with translation-invariant interactions and extend the harmonic-rigidity mechanism from the one-body setting to an interacting many-body problem. We prove a sharp upper bound on the ground-state center-of-mass fluctuation in terms of the active spectral gap associated with the center-of-mass probe, and show that this bound does not require any positivity assumption on the ground state. In the positivity class, we characterize the equality case completely. Exact saturation occurs if and only if the external one-body traps are harmonic with a common frequency, while the interaction may remain arbitrary within the translation-invariant class. We also identify a natural rigidity defect measuring deviation from the harmonic extremal situation and prove quantitative near-saturation estimates controlling both the variance deficit and the spectral weight outside the first active shell. In this way, the paper establishes harmonic confinement as the unique static extremizer for the rigid interacting center-of-mass mode at fixed active gap.

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