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Accelerating iterative eigenvalue algorithms is often achieved by employing aspectral shifting strategy. Unfortunately, improved shifting typically leads toa smaller eigenvalue for the resulting shifted operator, which in turn resultsin a high condition number of the underlying solution matrix, posing a majorchallenge for iterative linear solvers. This paper introduces a two-leveldomain decomposition preconditioner that addresses this issue for the linearSchr\"odinger eigenvalue problem, even in the presence of a vanishingeigenvalue gap in non-uniform, expanding domains. Since the quasi-optimalshift, 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 associatedeigenfunction as a generator to construct a coarse space. We analyze theresulting two-level additive Schwarz preconditioner and obtain a conditionnumber bound that is independent of the domain's anisotropy, despite the needfor only one basis function per subdomain for the coarse solver. Severalnumerical examples are presented to illustrate its flexibility and efficiency.

A Scalable Two-Level Domain Decomposition Eigensolver for Periodic Schr\"odinger Eigenstates in Anisotropically Expanding Domains