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This paper provides a provably quasi-optimal preconditioning strategy of thelinear Schr\"odinger eigenvalue problem with periodic potentials for a possiblynon-uniform spatial expansion of the domain. The quasi-optimality is achievedby having the iterative eigenvalue algorithms converge in a constant number ofiterations for different domain sizes. In the analysis, we derive an analyticfactorization of the spectrum and asymptotically describe it using conceptsfrom the homogenization theory. This decomposition allows us to express theeigenpair as an easy-to-calculate cell problem solution combined with anasymptotically vanishing remainder. We then prove that the easy-to-calculatelimit eigenvalue can be used in a shift-and-invert preconditioning strategy tobound the number of eigensolver iterations uniformly. Several numericalexamples illustrate the effectiveness of this quasi-optimal preconditioningstrategy.

A Quasi-Optimal Factorization Preconditioner for Periodic Schr\"odinger Eigenstates in Anisotropically Expanding Domains