Old and New Schrödinger Eigenvalue Localization

Unconditional guaranteed lower and upper eigenvalue bounds are mandatory for the understanding of the Schrödinger eigenvalue spectrum and its spectral gaps. While upper eigenvalue bounds are naturally induced by conforming discretizations, guaranteed lower eigenvalue bounds (GLB) are less immediate. This paper clarifies the adaptation of nonconforming GLB from the harmonic eigenvalue problem and discusses their comparison for general and piecewise constant potentials. A fine-tuned extra-stabilized scheme is proposed and found superior in numerical comparisons. This new direct calculation of GLB is compatible with adaptive mesh-refinement and successfully circumvents the appearance of maximal mesh-size parameters in former GLB based on postprocessing. Computational benchmarks also investigate guaranteed upper eigenvalue bounds (GUB) for two-sided eigenvalue control by conforming test functions associated to the underlying nonconforming computations. A numerical comparison with GUB from additional lowest-order conforming finite element schemes shows competitive accuracy with less computational cost.