open_problem: ax-2607.07851-15e7560c

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ax-2607.07851-15e7560c Problem 3.17 (Symplectic Schur–Horn problem for within-DOF uncertainties): Characterize, for fixed symplectic spectrum $\nu_{1}\geq\cdots\geq\nu_{n}>0$ , the attainable set $\mathcal{U}(\nu)=\Bigl\{\bigl(u_{1}(\Sigma),\dots,u_{n}(\Sigma)\bigr):\Sigma\in\text{the }\operatorname{Sp}(2n,\mathbb{R})\text{-orbit with spectrum }\nu\Bigr\}\subset\mathbb{R}_{>0}^{n},$ where $u_{j}(\Sigma)=\sqrt{\det\Sigma_{jj}}$ . In particular, we need to explore (a) is $\min_{j}u_{j}\geq\nu_{n}$ on the whole orbit (so that no DOF can be squeezed below the smallest symplectic eigenvalue)? (b) in the equipartitioned case $\nu_{j}\equiv\nu$ , is $u_{j}\geq\nu$ for every $j$ (the per-DOF form of the conjecture in )? And (c) describe $\mathcal{U}(\nu)$ by majorization-type inequalities, in analogy with the Schur–Horn theorem, using the symplectic eigenvalue technology of . Some partial cues include (i) $\prod_{j}u_{j}\geq\prod_{j}\nu_{j}$ (Theorem 3.16 ); (ii) $u_{j}\geq\nu_{n}$ would follow from the interlacing-type bound “every $2\times 2$ symplectic compression of $\Sigma$ has symplectic eigenvalue $\geq\nu_{n}$ ,” a statement of exactly the kind studied in ; and (iii) for $n=1$ , $\mathcal{U}(\nu)=\{\nu\}$ (Theorem 3.16 ). mathematical physics / math-ph enumerated in arXiv paper titled 'open problems' (formal problem environment): Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invarian https://arxiv.org/abs/2607.07851   open   ts-synth unclassified 2026-09-02T17:57:21Z 2026-09-02T17:57:21Z

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