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⊕(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.Comment: Proceedings of the London Mathematical Society (2009) in pres