谱污染及其避免方法（在狄拉克和周期薛定谔算子中的应用）
Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
AI总结：
本文研究谱污染问题，提出基于正交投影的伽辽金基方法，并应用于周期薛定谔算子和狄拉克算子，证明在特定基下可避免谱污染。
摘要：
本文致力于研究谱污染，包含抽象结果及其在量子力学中出现的本质谱有间隙的自伴算子上的应用。首先，我们考虑伽辽金基，该基尊重由固定正交投影$P$给出的环境希尔伯特空间直和分解$H=PH\oplus(1-P)H$，并精确确定污染谱的位置。随后应用于周期薛定谔算子（在Wannier型基中无污染）和狄拉克算子（考虑了几种自然分解）。在第二部分中，我们添加约束条件，即在伽辽金基中，$PH$中的向量与$(1-P)H$中的向量之间存在某种关系。证明了抽象结果，并将其应用于几种实用方法，如相对论量子力学中著名的“动能平衡”方法。
英文摘要：
This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some self-adjoint operators with a gap in their essential spectrum occuring in Quantum Mechanics. First we consider Galerkin basis which respect the decomposition of the ambient Hilbert space into a direct sum $H=PH\oplus(1-P)H$, given by a fixed orthogonal projector $P$, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schrödinger operators (pollution is absent in a Wannier-type basis), and to Dirac operator (several natural decompositions are considered). In the second part, we add the constraint that within the Galerkin basis there is a certain relation between vectors in $PH$ and vectors in $(1-P)H$. Abstract results are proved and applied to several practical methods like the famous "kinetic balance" of relativistic Quantum Mechanics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
