Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
Lewin, Mathieu ; Séré, Eric
HAL, hal-00346352 / Harvested from HAL
Text on HAL
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.
Publié le : 2010-07-05
Classification:  spectral pollution,  spurious eigenvalues,  variational collapse,  periodic Schrödinger operators,  Wannier functions,  Dirac operator,  kinetic balance method,  [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP],  [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph],  [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph],  [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] 
@article{hal-00346352,
     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 = {HAL},
     volume = {2010},
     number = {0},
     year = {2010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-00346352}
}

Lewin, Mathieu; Séré, Eric. Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators). HAL, Tome 2010 (2010) no. 0, . http://gdmltest.u-ga.fr/item/hal-00346352/