发表机构
University of Haifa（海法大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究量子力学自举法的紧性，发现平稳自举法存在无法满足物理态正性的反例，本征态自举法在多数测试中无违反，推测其对动量矩隐含阿基米德型界并提出紧性陈述。
AI 中文摘要
量子力学的数值自举法仅针对平方和测试候选态的正性，而每个物理态对每个逐点非负多项式都赋予非负期望值。在一维中两者一致；在二维及更高维度中则不一致。该差距是否存在很大程度上取决于所施加的约束集。对于平稳自举法（施加⟨[H,O]⟩=0，是适用于热态和混合态的松弛方法），我们展示了一个二维四次双势阱和一个矩向量，其精确满足所有三阶平稳约束，具有正定矩矩阵，但对R²上非负的多项式赋予负期望值；因此它不是任何态的矩序列。所有数据均为有理数，每一步都通过精确算术验证。对于本征态自举法（额外施加⟨OH⟩=E⟨O⟩），相同搜索在五个设置中均未发现违反情况，这些设置涵盖二维、三维和五自由度，截断水平为三阶和四阶，针对完全分离多项式族进行测试。唯一例外出现在截断水平极低、仅保留一个本征态约束的情况，本文提出的理论可解释该例外。我们确定了机制：本征态约束约束了高动量矩，而这些矩在可行集上原本无界，正是这些无界方向到达了两个锥之间的区域。最后，在典型势的反射对称性下，相关阻碍是余正性而非非负性，对于最低截断水平下可用的四次见证，首次可能的失败出现在五自由度。我们推测本征态约束对动量矩隐含了阿基米德型界，并提出相应的紧性陈述。
英文摘要
The numerical bootstrap for quantum mechanics tests a candidate state's positivity only against sums of squares, whereas every physical state assigns nonnegative expectation to every pointwise nonnegative polynomial. In one dimension the two coincide; in two or more they do not. Whether this gap is realized depends sharply on which constraint set is imposed. For the stationary bootstrap, which imposes <[H,O]> = 0 and is the relaxation appropriate to thermal and mixed states, we exhibit a two-dimensional quartic double well and a moment vector that satisfies every level-three stationary constraint exactly, has a positive definite moment matrix, and yet assigns a negative expectation to a polynomial nonnegative on R^2; it is therefore the moment sequence of no state. All data are rational, and every step is verified in exact arithmetic. For the eigenstate bootstrap, which additionally imposes <OH> = E<O>, the same search finds no violation in any of five settings spanning two, three and five degrees of freedom and truncation levels three and four, tested against complete families of separating polynomials. The single exception occurs at a truncation so low that only one eigenstate constraint survives, and the theory presented here accounts for it. We identify the mechanism: the eigenstate constraints bound the high momentum moments, otherwise unbounded on the feasible set, and it is those unbounded directions that reach the region between the two cones. Finally, under the reflection symmetries of a typical potential the relevant obstruction is copositivity rather than nonnegativity, which, for the quartic witnesses available at the lowest truncation, places the first possible failure at five degrees of freedom. We conjecture that the eigenstate constraints imply an Archimedean-type bound on the momentum moments, and formulate the corresponding tightness statement.
Commentsv3: corrects the degree range in Conjecture 1 to k <= K - (deg V)/2 + 1; the range stated in v1 and v2 is the quartic case. Adds a remark that when deg V = 2K the eigenstate constraints retain nothing beyond the energy, with a counterexample due to Ye Zhou. Theorem 1, Proposition 1, and all numerics are unchanged
