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.Comment: 30 pages, 7 figures, 2 table