Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
For researchers in numerical analysis and quantum mechanics, this work offers a rigorous understanding of spectral pollution and practical guidance to avoid it, though the results are theoretical and not yet demonstrated on large-scale problems.
This paper provides a theoretical framework to understand and avoid spectral pollution in Galerkin approximations of self-adjoint operators with spectral gaps, and applies it to periodic Schrödinger and Dirac operators, showing that pollution can be eliminated with appropriate basis choices.
This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some self-adjoint operators with a gap in their essential spectrum occuring in Quantum Mechanics. First we consider Galerkin basis which respect the decomposition of the ambient Hilbert space into a direct sum $H=PH\oplus(1-P)H$, given by a fixed orthogonal projector $P$, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schrödinger operators (pollution is absent in a Wannier-type basis), and to Dirac operator (several natural decompositions are considered). In the second part, we add the constraint that within the Galerkin basis there is a certain relation between vectors in $PH$ and vectors in $(1-P)H$. Abstract results are proved and applied to several practical methods like the famous "kinetic balance" of relativistic Quantum Mechanics.