Old and new Schr\"odinger eigenvalue localisation

TL;DR

This paper advances the computation of guaranteed lower and upper bounds for Schrödinger eigenvalues, introducing a new stabilised scheme and comparing its performance with existing methods.

Contribution

It adapts nonconforming eigenvalue bounds to Schrödinger problems, proposes a superior stabilised scheme, and demonstrates its effectiveness through numerical benchmarks.

Findings

The new stabilised scheme outperforms previous methods in accuracy.

The approach is compatible with adaptive mesh refinement.

Guaranteed bounds are achieved with reduced computational cost.

Abstract

Unconditional guaranteed lower and upper eigenvalue bounds are mandatory for the understanding of the Schr\"odinger 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.