Spectral pollution and how to avoid it

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, occurring in Quantum Mechanics. First we consider Galerkin bases which preserve 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 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 Pℌ and vectors in (1−P)ℌ. Abstract results are proved and applied to several practical methods like the famous kinetic balance condition of relativistic Quantum Mechanics.