High-order guaranteed lower bounds for the Schrödinger eigenvalue problem

Abstract

We compute unconditional guaranteed lower eigenvalue bounds for Schrödinger operators using a modified hybrid high-order method with built-in correction for the potential defect. The method assumes exact numerical integration and allows possibly unbounded potentials in $L^p$ for $p>n/2$ and $p\geq2$ on general polyhedral domains. We prove that the high-order correction preserves optimal a priori convergence orders, and derive reliable and efficient a posteriori error estimators. Numerical experiments provide striking evidence of optimal high-order convergence rates of the adaptive algorithm.