This paper provides a provably quasi-optimal preconditioning strategy of the linear Schrödinger eigenvalue problem with periodic potentials for a possibly nonuniform 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.


  1. periodic Schrödinger equation
  2. iterative eigenvalue solvers
  3. preconditioner
  4. asymptotic eigenvalue analysis
  5. factorization principle
  6. directional homogenization

MSC codes

  1. 65N25
  2. 65F15
  3. 65N30
  4. 35B27
  5. 35B40

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2508 - 2537
ISSN (online): 1095-7170


Submitted: 3 November 2021
Accepted: 1 June 2022
Published online: 15 September 2022


  1. periodic Schrödinger equation
  2. iterative eigenvalue solvers
  3. preconditioner
  4. asymptotic eigenvalue analysis
  5. factorization principle
  6. directional homogenization

MSC codes

  1. 65N25
  2. 65F15
  3. 65N30
  4. 35B27
  5. 35B40



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 411724963

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