标题： 有限维逼近对于希尔伯特空间算子及其在量子力学中的应用 显示英文标题

标题： Finite dimensional approximations for Hilbert space operators and applications in Quantum Mechanics

摘要： 在本工作中，我们开发了一个统一的框架，用于希尔伯特空间上线性算子的拟对角和F\o lner型逼近。 这些逼近（最初为有界算子和算子代数制定）涉及非零有限秩正交投影序列，这些序列在渐近意义上与算子相容——在范数下（拟对角）或在均值下（F\o lner）。 这种结构保证了它们的有限部分的谱逼近结果。 我们将这一理论扩展到无界、稠密定义的可闭算子，建立了Halmos经典结果的推广：每个可闭的拟对角算子是同一定义域上可闭块对角算子的紧扰动。 同样，我们引入稀疏F\o lner序列，并建立拟对角逼近与稀疏F\o lner序列存在之间的相互作用。 理论发展通过使用不同类型加权移位的显式例子进行说明，并应用于量子力学模型，包括Weyl代数及其Schrödinger表示的详细处理。 显示英文摘要

摘要： In this work, we develop a unified framework for quasidiagonal and F\o lner-type approximations of linear operators on Hilbert spaces. These approximations (originally formulated for bounded operators and operator algebras) involve sequences of non-zero finite rank orthogonal projections that asymptotically commute with the operator -- either in norm (quasidiagonal) or in mean (F\o lner). Such structures guarantee spectral approximation results in terms of their finite sections. We extend this theory to unbounded, densely defined closable operators, establishing a generalization of Halmos' classical result: every closable quasidiagonal operator is a compact perturbation of a closable block-diagonal operator on the same domain. Likewise, we introduce sparse F\o lner sequences and establish an interplay between quasidiagonal approximations and the existence of sparse F\o lner sequences. The theoretical developments are illustrated with explicit examples using different types of weighted shifts and applied to quantum mechanical models, including a detailed treatment of the Weyl algebra and its Schr\"odinger representation.