  This paper provides a provably quasi-optimal preconditioning strategy of the
linear Schr\"odinger eigenvalue problem with periodic potentials for a possibly
non-uniform spatial expansion of the domain. The quasi-optimality is achieved
by having the iterative eigenvalue algorithms converge in a constant number of
iterations for 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
bound the number of eigensolver iterations uniformly. Several numerical
examples illustrate the effectiveness of this quasi-optimal preconditioning
strategy.
