Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schr\"odinger Operators)

TL;DR

This paper investigates spectral pollution in self-adjoint operators, providing abstract results and practical applications to quantum mechanical operators, and proposes methods to avoid pollution in computational spectral analysis.

Contribution

It introduces a framework for localizing spectral pollution and applies it to Dirac and periodic Schrödinger operators, including the analysis of pollution-free bases and practical methods like kinetic balance.

Findings

Pollution is absent in Wannier-type bases for periodic Schrödinger operators.

Spectral pollution can be precisely localized using the proposed methods.

The kinetic balance method effectively reduces spectral pollution in relativistic quantum computations.

Abstract

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 Schr\"odinger 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.