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
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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
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