发表机构
Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在Smale计算纲领下，证明一维薛定谔算子谱型分解的尖锐计算界：纯点和绝对连续谱需一个极限，奇异连续谱需两个；并用小波认证计算给出匹配上界，确立谱类型的尖锐层级。
AI 中文摘要
我们证明了在Smale计算基础纲领精神下确定薛定谔算子谱型分解的尖锐界。对于$L^2(\mathbb R)$上显式的一维自伴薛定谔算子$H=-{\mathrm d^2}/{\mathrm dx^2}+V$，其中$V\in C^\infty(\mathbb R;\mathbb R)$由所有导数及其界的有限描述给出，纯点和绝对连续谱集通常无法通过任何单一极限过程恢复。奇异连续谱集严格更难：它通常无法通过两个嵌套极限过程恢复。二分法的解析构造实现了下界：纯点谱的Gordon型重复、绝对连续谱的高势垒，以及基于Riesz乘积、矩消减扰动和计算Gelfand--Levitan方案的奇异连续谱的逆谱构造。有限描述框架也暗示了在固定形式系统中（例如在计算机辅助证明中使用时）可被认证内容的相应限制。相反，使用基于小波的认证计算，我们证明了$\mathbb R^d$上具有局部有界变差系数和定量局部变差控制的广泛自伴微分算子类的匹配上界：两个极限足以处理纯点和绝对连续部分，三个极限足以处理奇异连续部分。这为谱类型提供了尖锐的层级结构。
英文摘要
We prove sharp bounds for determining the spectral-type decomposition of Schrödinger operators in the spirit of Smale's program on the foundations of computation. For explicit one-dimensional self-adjoint Schrödinger operators $H=-{\mathrm d^2}/{\mathrm dx^2}+V$ on $L^2(\mathbb R)$, where $V\in C^\infty(\mathbb R;\mathbb R)$ is given by a finite description of all derivatives and derivative bounds, the pure point and absolutely continuous spectral sets cannot, in general, be recovered by any single limiting procedure. The singular continuous spectral set is strictly harder: it cannot, in general, be recovered by two nested limiting procedures. Analytic constructions of dichotomies realize the lower bounds: Gordon-type repetitions for pure point spectrum, high barriers for absolutely continuous spectrum, and an inverse spectral construction for singular continuous spectrum based on Riesz products, moment-killing perturbations, and a computational Gelfand--Levitan scheme. The finite-description framework also implies corresponding limitations on what can be certified in fixed formal systems (e.g., when used in computer-assisted proofs). Conversely, using wavelet-based certified computation, we prove matching upper bounds for broad classes of self-adjoint differential operators on $\mathbb R^d$ with coefficients of locally bounded variation and quantitative local variation control: two limits suffice for the pure point and absolutely continuous parts, and three for the singular continuous part. This provides a sharp hierarchy for spectral types.
