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

TL;DR

This paper introduces a quasi-optimal preconditioning method for the Schrödinger eigenproblem with periodic potentials, ensuring constant iteration convergence across varying domain sizes by spectral factorization and homogenization insights.

Contribution

It develops a new preconditioning strategy based on spectral factorization and homogenization theory, achieving uniform convergence for eigenvalue algorithms in anisotropically expanding domains.

Findings

Iterative eigenvalue algorithms converge in a constant number of iterations.

The preconditioner is effective for non-uniform, expanding domains.

Numerical examples confirm the theoretical results.

Abstract

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.