Old and new Schrödinger eigenvalue localisation
For researchers in numerical analysis and computational quantum mechanics, this work offers an improved method for guaranteed eigenvalue bounds, though it is an incremental improvement over existing techniques.
The paper proposes a new direct calculation of guaranteed lower eigenvalue bounds for Schrödinger eigenvalue problems, which avoids maximal mesh-size parameters and is compatible with adaptive mesh-refinement. Numerical comparisons show it is superior to existing methods and provides competitive accuracy for two-sided eigenvalue control at lower computational cost.
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 discretisations, 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-stabilised 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 post-processing. 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.