Spectral pollution and how to avoid it

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, occurring in Quantum Mechanics.First we consider Galerkin bases which preserve the decomposition of the ambient Hilbert space into a direct sum = P⊕(1-P) given by a fixed orthogonal projector P, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schrödinger operators (we show that pollution is absent in a Wannier-type basis) and to Dirac operators (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 P and vectors in (1-P). Abstract results are proved and applied to several practical methods like the famous kinetic balance condition of relativistic Quantum Mechanics.