Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schr\"odinger Operators)
Lewin, Mathieu ; Séré, Eric
arXiv, 0812.2153 / Harvested from arXiv
Text on arXiv
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 Schr\"odinger 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.
Publié le : 2008-12-11
Classification:  Mathematics - Spectral Theory,  Mathematical Physics,  Mathematics - Numerical Analysis 
@article{0812.2153,
     author = {Lewin, Mathieu and S\'er\'e, Eric},
     title = {Spectral Pollution and How to Avoid It (With Applications to Dirac and
  Periodic Schr\"odinger Operators)},
     journal = {arXiv},
     volume = {2008},
     number = {0},
     year = {2008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0812.2153}
}

Lewin, Mathieu; Séré, Eric. Spectral Pollution and How to Avoid It (With Applications to Dirac and
  Periodic Schr\"odinger Operators). arXiv, Tome 2008 (2008) no. 0, . http://gdmltest.u-ga.fr/item/0812.2153/