Journal of Spectral Theory


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Volume 2, Issue 4, 2012, pp. 329–354
DOI: 10.4171/JST/32

Published online: 2012-06-24

Generalized Weyl theorems and spectral pollution in the Galerkin method

Lyonell Boulton[1], Nabile Boussaïd[2] and Mathieu Lewin[3]

(1) Heriot-Watt University, Edinburgh, UK
(2) Université de Franche-Comté, Besançon, France
(3) Université de Cergy-Pontoise, France

We consider a general framework for investigating spectral pollution in the Galerkin method. We show how this phenomenon is characterised via the existence of particular Weyl sequences which are singular in a suitable sense. For a given semi-bounded selfadjoint operator A, we identify relative compactness conditions on a selfadjoint perturbation B, ensuring that the limiting set of spectral pollution of A and B coincide. Our results show that, under perturbation, this limiting set behaves in a similar fashion as the essential spectrum.

Keywords: Weyl's Theorem, generalized essential spectrum, spectral pollution, Galerkin method

Boulton Lyonell, Boussaïd Nabile, Lewin Mathieu: Generalized Weyl theorems and spectral pollution in the Galerkin method. J. Spectr. Theory 2 (2012), 329-354. doi: 10.4171/JST/32