Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schr\"odinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.

翻译：通过采用谱位移策略来加速迭代特征值算法是常见方法。然而，改进的位移通常会导致位移算子的特征值减小，进而引发底层求解矩阵的高条件数，这给迭代线性求解器带来了重大挑战。本文针对线性薛定谔特征值问题，提出了一种双层区域分解预处理器，即使在非均匀膨胀域中出现特征值间隙消失的情况下也能解决该问题。由于特征值求解器需要准最优位移（该位移已作为谱单元问题的解存在），因此自然可以利用其关联的特征函数作为生成元来构建粗空间。我们分析了由此产生的双层加性施瓦茨预处理器，并得到了一个独立于域各向异性的条件数界，尽管粗求解器每个子域仅需一个基函数。通过多个数值算例展示了该方法的灵活性与高效性。