Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)

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 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.

Proceedings of the London Mathematical Society (2009) in press

References in corpus (4)

• Exponential localization of Wannier functions in insulators
• On the eigenvalues of operators with gaps. Application to Dirac operators
• Self-adjointness of Dirac operators via Hardy-Dirac inequalities
• Non-perturbative embedding of local defects in crystalline materials

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