Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
Lewin, Mathieu; Séré, Eric (2009), Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators), Proceedings of the London Mathematical Society, 100, 3, p. 864-900. http://dx.doi.org/10.1112/plms/pdp046
TypeArticle accepté pour publication ou publié
Journal nameProceedings of the London Mathematical Society
MetadataShow full item record
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.
Subjects / Keywordsvariational collapse; periodic Schrödinger operators; Wannier functions; Dirac operator; kinetic balance method; spectral pollution; spurious eigenvalues
Showing items related by title and author.
General results on the eigenvalues of operators with gaps, arising from both ends of the gaps. Application to Dirac operators. Dolbeault, Jean; Esteban, Maria J.; Séré, Eric (2006) Article accepté pour publication ou publié