发表机构
Department of Computer Science, University of Haifa（海法大学计算机系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明本征态量子自举在有限阶非精确下仍可收敛至物理矩集，以六次谐振子为例，并推广至多项式薛定谔算子及四模扩展，揭示有限阶可行点无正测度的现象。
AI 中文摘要
本征态量子力学自举可以在保持每一有限阶非精确性的同时收敛到物理矩集。我们在一维准精确可解六次谐振子的显式已知本征值（包括激发态）上证明了这一点。在每一有限阶，$u^2$的下界和上界严格低于和高于其物理值，即使附加任意有限多项式Gram检验以及保留矩空间中可见的所有本征态湮灭关系也是如此。因此，中心可观测量的任一侧都不存在模本征态湮灭子的有限平方和证书。在权重在多项式向量上严格为正的任意有限附加局部化约束下，该间隙持续存在。然后，我们证明了满足显式平方和约束和导数界条件的多项式薛定谔算子的收敛性。本征态关系给出了截断阶一致的矩估计；这些估计产生一个在指定本征空间上支撑的正则薛定谔表示。最后，一个四模扩展将该固定可观测量的障碍与每一有限阶没有正表示测度的可行位置矩相结合。在第三阶，该构造产生了具有严格正定矩矩阵的非物理可行点。
英文摘要
The eigenstate quantum-mechanical bootstrap can converge to the physical moment set while remaining nonexact at every finite order. We prove this for a one-dimensional quasi-exactly solvable sextic oscillator at explicitly known eigenvalues, including excited levels. At every finite order, the lower and upper bounds on $u^2$ remain strictly below and above its physical value, even after adjoining arbitrary finite polynomial Gram tests and all eigenstate-annihilator relations visible in the retained moment space. Hence neither side of the centered observable admits a finite sum-of-squares certificate modulo the eigenstate annihilator. The gap persists under any finite set of additional localizing constraints whose weights are strictly positive on polynomial vectors. We then prove convergence for polynomial Schrödinger operators satisfying explicit sum-of-squares confinement and derivative bounds. The eigenstate relations give moment estimates uniform in the truncation order; these yield a regular Schrödinger representation supported on the prescribed eigenspace. Finally, a four-mode extension combines this fixed-observable obstruction with feasible position moments having no positive representing measure at every finite order. At level three, this construction yields nonphysical feasible points with strictly positive definite moment matrices.
Comments36 pages. Numerical scripts and solver inputs and outputs included as ancillary files. Companion paper: arXiv:2610.03362
