no. 7
Elementary linear algebra for advanced spectral problems
[Algèbre linéaire élémentaire pour des problèmes d’analyse spectrale]
Sjöstrand, Johannes1 ; Zworski, Maciej2
1 École Polytechnique Centre de Mathématiques Laurent Schwartz UMR 7460, CNRS 91128 Palaiseau (France)
2 University of California Mathematics Department Evans Hall Berkeley, CA 94720 (USA)
Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2095-2141.

Nous décrivons une idée simple d’algèbre linéaire, qui a été utilisée dans différentes branches des mathématiques, telles que la théorie des bifurcations, les équations aux dérivées partielles et l’analyse numérique. Sous le nom de la méthode des compléments de Schur c’est un des outils standard de l’algèbre linéaire appliquée. En e.d.p. et en analyse spectrale elle est parfois appelée la méthode des problèmes de Grushin, et ici nous nous concentrons sur son utilisation dans l’étude des problèmes en dimension infinie, venant des équations aux dérivées partielles de la physique mathématique.

We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators of mathematical physics.

DOI : 10.5802/aif.2328
Classification : 15A21, 35P05, 35Q40, 81Q15
Keywords: Grushin problem, Schur complement, Feshbach reduction, eigenvalues, resonances, trace formulæ
Mot clés : problème de Grushin, complément de Schur, réduction de Feschbach, valeurs propres, résonances, formules de trace
Sjöstrand, Johannes 1 ; Zworski, Maciej 2

1 École Polytechnique Centre de Mathématiques Laurent Schwartz UMR 7460, CNRS 91128 Palaiseau (France)
2 University of California Mathematics Department Evans Hall Berkeley, CA 94720 (USA) 
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Sjöstrand, Johannes; Zworski, Maciej. Elementary linear algebra for advanced spectral problems. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2095-2141. doi : 10.5802/aif.2328. http://www.numdam.org/articles/10.5802/aif.2328/

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