从估计到证明：奇异薛定谔算子的基态能量的可验证边界
From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators
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中文总结 AI 辅助
该研究提出计算框架，为奇异薛定谔算子的基态能量返回数学可验证边界，首次实现氢分子离子真实无限域基态能量的可保证双侧包围，且精度损失极小，将特征值计算从估计转为证明。
中文摘要 AI 辅助
量子化学、材料科学和谱几何中的许多预测都依赖于薛定谔算子的最低能级，但标准模拟返回的近似值无法保证它们与真实值的差距。我们提出了一种计算框架，该框架能返回数学上可验证的边界——一个必然包含精确能量的区间，即使对于现有可验证方法无法处理的真实分子的奇异、无界、变号势也适用。对于氢分子离子，该框架提供了据我们所知首个对真实无限域基态能量的可保证双侧包围：宽度小于5×10⁻⁴的区间，必然包含公认的参考值。使全空间保证成为可能的关键是对将问题限制在有限盒子中的误差的显式、可计算边界——替代了仅假设波函数衰减的经典论证——该边界从两侧包围真实能量。值得注意的是，该可验证区间将未验证的数值结果重现到了11位，因此数学严谨性几乎未在精度上付出代价。该成果将特征值计算从估计转变为证明。
英文摘要
Many predictions in quantum chemistry, materials science, and spectral geometry hinge on the lowest energy levels of a Schrödinger operator, yet standard simulations return approximations with no guarantee of how far they sit from the true value. We present a computational framework that returns mathematically certified bounds---an interval provably containing the exact energy---even for the singular, unbounded, sign-changing potentials of real molecules, where existing certified methods fail. For the hydrogen molecular ion this framework delivers, to our knowledge, the first guaranteed two-sided enclosure of the true infinite-domain ground-state energy: an interval of width below $5\times10^{-4}$ that provably contains the accepted reference value. The key that makes a whole-space guarantee possible is an \emph{explicit}, computable bound on the error of restricting the problem to a finite box---replacing the classical argument that the wavefunction merely decays---which brackets the true energy from both sides. Remarkably, the certified interval reproduces the uncertified numerical value to eleven digits, so mathematical rigor costs almost nothing in accuracy. The result turns eigenvalue computation from an estimate into a proof.
