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) 
@article{AIF_2007__57_7_2095_0,
     author = {Sj\"ostrand, Johannes and Zworski, Maciej},
     title = {Elementary linear algebra for advanced spectral problems},
     journal = {Annales de l'Institut Fourier},
     pages = {2095--2141},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     doi = {10.5802/aif.2328},
     mrnumber = {2394537},
     zbl = {1140.15009},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2328/}
}
TY  - JOUR
AU  - Sjöstrand, Johannes
AU  - Zworski, Maciej
TI  - Elementary linear algebra for advanced spectral problems
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 2095
EP  - 2141
VL  - 57
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2328/
DO  - 10.5802/aif.2328
LA  - en
ID  - AIF_2007__57_7_2095_0
ER  - 
%0 Journal Article
%A Sjöstrand, Johannes
%A Zworski, Maciej
%T Elementary linear algebra for advanced spectral problems
%J Annales de l'Institut Fourier
%D 2007
%P 2095-2141
%V 57
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2328/
%R 10.5802/aif.2328
%G en
%F AIF_2007__57_7_2095_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2328/

[1] Bau, David; Trefethen, Lloyd N. Numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997 | MR | Zbl

[2] Boutet de Monvel, Louis Boundary problems for pseudo-differential operators, Acta Math., Volume 126 (1971) no. 1-2, pp. 11-51 | DOI | MR | Zbl

[3] Davies, E.B.; Hager, M. Perturbations of Jordan matrices (arXiv:math/0612158v)

[4] Dereziński, Jan; Jakšić, Vojkan Spectral theory of Pauli-Fierz operators, J. Funct. Anal., Volume 180 (2001) no. 2, pp. 243-327 | DOI | MR | Zbl

[5] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[6] Fletcher, Roger; Johnson, Tom On the stability of null-space methods for KKT systems, SIAM J. Matrix Anal. Appl., Volume 18 (1997) no. 4, pp. 938-958 | DOI | MR | Zbl

[7] Gohberg, I. C.; Kreĭn, M. G. Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969 | MR | Zbl

[8] Grushin, V. V. Les problèmes aux limites dégénérés et les opérateurs pseudo-différentiels, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 737-743 | MR | Zbl

[9] Hager, M.; Sjöstrand, J. Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators (arXiv: math.SP/0601381)

[10] Helffer, B.; Sjöstrand, J. Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) (1986) no. 24-25, pp. iv+228 | EuDML | Numdam | MR | Zbl

[11] Helffer, B.; Sjöstrand, J. Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988) (Lecture Notes in Phys.), Volume 345, Springer, Berlin, 1989, pp. 118-197 | MR | Zbl

[12] Helffer, B.; Sjöstrand, J. Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mém. Soc. Math. France (N.S.) (1989) no. 39, pp. 1-124 | EuDML | Numdam | Zbl

[13] Hörmander, Lars The analysis of linear partial differential operators. I, II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256, 257, Springer-Verlag, Berlin, 1983 (Distribution theory and Fourier analysis) | MR | Zbl

[14] Hörmander, Lars The analysis of linear partial differential operators. III, IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274,275, Springer-Verlag, Berlin, 1985 | MR | Zbl

[15] Iantchenko, A.; Sjöstrand, J.; Zworski, M. Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett., Volume 9 (2002) no. 2-3, pp. 337-362 | MR | Zbl

[16] Lidskiĭ, V. B. Perturbation theory of non-conjugate operators, U.S.S.R. Comput. Math. and Math. Phys., Volume 6 (1966), pp. 73-85 | DOI | Zbl

[17] Melin, Anders; Sjöstrand, Johannes Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Astérisque (2003) no. 284, pp. 181-244 (Autour de l’analyse microlocale) | MR | Zbl

[18] Moro, Julio; Burke, James V.; Overton, Michael L. On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure, SIAM J. Matrix Anal. Appl., Volume 18 (1997) no. 4, pp. 793-817 | DOI | MR | Zbl

[19] Sjöstrand, Johannes Operators of principal type with interior boundary conditions, Acta Math., Volume 130 (1973), pp. 1-51 | DOI | MR | Zbl

[20] Sjöstrand, Johannes Pseudospectrum for differential operators, Seminaire: Équations aux Dérivées Partielles, 2002–2003 (Sémin. Équ. Dériv. Partielles), École Polytech., Palaiseau, 2003, pp. Exp. No. XVI, 9 | EuDML | Numdam | MR | Zbl

[21] Sjöstrand, Johannes; Vodev, Georgi Asymptotics of the number of Rayleigh resonances, Math. Ann., Volume 309 (1997) no. 2, pp. 287-306 (With an appendix by Jean Lannes) | DOI | MR | Zbl

[22] Sjöstrand, Johannes; Zworski, Maciej Asymptotic distribution of resonances for convex obstacles, Acta Math., Volume 183 (1999) no. 2, pp. 191-253 | DOI | MR | Zbl

[23] Sjöstrand, Johannes; Zworski, Maciej Quantum monodromy and semi-classical trace formulae, J. Math. Pures Appl. (9), Volume 81 (2002) no. 1, pp. 1-33 (See also Quantum monodromy revisited, math.berkeley.edu/~zworski/qmr.ps) | MR | Zbl

[24] Trefethen, Lloyd N. Pseudospectra of linear operators, SIAM Rev., Volume 39 (1997) no. 3, pp. 383-406 | DOI | MR | Zbl

[25] Zelditch, S. Survey on the inverse spectral problem (to appear) | Zbl

[26] Zworski, Maciej Resonances in physics and geometry, Notices of the AMS, Volume 46 (1999) no. 3, pp. 319-328 | MR | Zbl

[27] Zworski, Maciej Numerical linear algebra and solvability of partial differential equations, Comm. Math. Phys., Volume 229 (2002) no. 2, pp. 293-307 | DOI | MR | Zbl

Cité par Sources :