This paper provides a provably optimal preconditioning strategy of the linear Schr\"odinger eigenvalue problem with periodic potentials for a possibly non-uniform spatial expansion of the domain. The optimality is achieved by having the iterative eigenvalue algorithms converge in a constant number of iterations with respect to different domain sizes. In the analysis, we derive an analytic factorization of the spectrum and asymptotically describe it using concepts from the homogenization theory. This decomposition allows us to express the eigenpair as an easy-to-calculate cell problem solution combined with an asymptotically vanishing remainder. We then prove that the easy-to-calculate limit eigenvalue can be used in a shift-and-invert preconditioning strategy to uniformly bound the number of eigensolver iterations. Several numerical examples illustrate the effectiveness of this optimal preconditioning strategy.

翻译：本文为线性Schr\'odinger egenvaly 问题提供了一个可以确定的最佳先决条件战略, 其周期性潜力有可能使域内空间扩张不统一。 最佳性是通过迭代的 egenvaly 算法在不同域大小的迭代中以恒定数趋同来实现的。 在分析中, 我们用同质理论的概念对频谱进行分析性系数化, 并用无源描述它。 这种分解使我们能够将igenpair 表达为容易计算细胞问题的一种解决方案, 并结合一种无源消散的剩余部分。 然后, 我们证明, 容易计算的限制 egenvality 可用于一个转变和反向的前提性战略, 以一致地将eigensoolver 的外观数捆绑在一起。 几个数字例子说明了这一最佳的前提条件战略的有效性 。