Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
Abstract
This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some selfadjoint 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 $H=PH\oplus(1P)H$, 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 Wanniertype 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 $(1P)H$. Abstract results are proved and applied to several practical methods like the famous "kinetic balance" of relativistic Quantum Mechanics.
 Publication:

arXiv eprints
 Pub Date:
 December 2008
 arXiv:
 arXiv:0812.2153
 Bibcode:
 2008arXiv0812.2153L
 Keywords:

 Mathematics  Spectral Theory;
 Mathematical Physics;
 Mathematics  Numerical Analysis
 EPrint:
 Proceedings of the London Mathematical Society (2009) in press