Date
2009
Dewey
Probabilités et mathématiques appliquées
Sujet
variational collapse; periodic Schrödinger operators; Wannier functions; Dirac operator; kinetic balance method; spectral pollution; spurious eigenvalues
Journal issue
Proceedings of the London Mathematical Society
Volume
100
Number
3
Publication date
12-2009
Article pages
864-900
Publisher
London Mathematical Society
Author
Lewin, Mathieu
Séré, Eric
Type
Article accepté pour publication ou publié
Abstract (EN)
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.