Elementary linear algebra for advanced spectral problems
[Algèbre linéaire élémentaire pour des problèmes d’analyse spectrale]
Sjöstrand, Johannes ; Zworski, Maciej
Annales de l'Institut Fourier, Tome 57 (2007), p. 2095-2141 / Harvested from Numdam
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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.

Publié le : 2007-01-01
DOI : https://doi.org/10.5802/aif.2328
Classification:  15A21,  35P05,  35Q40,  81Q15
Mots clés: problème de Grushin, complément de Schur, réduction de Feschbach, valeurs propres, résonances, formules de trace 
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     title = {Elementary linear algebra for advanced spectral problems},
     journal = {Annales de l'Institut Fourier},
     volume = {57},
     year = {2007},
     pages = {2095-2141},
     doi = {10.5802/aif.2328},
     zbl = {1140.15009},
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Sjöstrand, Johannes; Zworski, Maciej. Elementary linear algebra for advanced spectral problems. Annales de l'Institut Fourier, Tome 57 (2007) pp. 2095-2141. doi : 10.5802/aif.2328. http://gdmltest.u-ga.fr/item/AIF_2007__57_7_2095_0/

[1] Bau, David; Trefethen, Lloyd N. Numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1997) | MR 1444820 | Zbl 0874.65013

[2] Boutet De Monvel, Louis Boundary problems for pseudo-differential operators, Acta Math., Tome 126 (1971) no. 1-2, pp. 11-51 | Article | MR 407904 | Zbl 0206.39401

[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., Tome 180 (2001) no. 2, pp. 243-327 | Article | MR 1814991 | Zbl 1034.81016

[5] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 268 (1999) | MR 1735654 | Zbl 0926.35002

[6] Fletcher, Roger; Johnson, Tom On the stability of null-space methods for KKT systems, SIAM J. Matrix Anal. Appl., Tome 18 (1997) no. 4, pp. 938-958 | Article | MR 1472003 | Zbl 0890.65060

[7] Gohberg, I. C.; Kreĭn, M. G. Introduction to the theory of linear nonselfadjoint operators, American Mathematical Society, Providence, R.I., Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18 (1969) | MR 246142 | Zbl 0181.13504

[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 509187 | Zbl 0244.35075

[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 | Numdam | MR 871788 | Zbl 0631.35075

[11] Helffer, B.; Sjöstrand, J. Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988), Springer, Berlin (Lecture Notes in Phys.) Tome 345 (1989), pp. 118-197 | MR 1037319 | Zbl 0699.35189

[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 | Numdam | Zbl 0725.34099

[13] Hörmander, Lars The analysis of linear partial differential operators. I, II, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 256, 257 (1983) (Distribution theory and Fourier analysis) | MR 717035 | Zbl 0521.35001

[14] Hörmander, Lars The analysis of linear partial differential operators. III, IV, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 274,275 (1985) | MR 781536 | Zbl 0601.35001

[15] Iantchenko, A.; Sjöstrand, J.; Zworski, M. Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett., Tome 9 (2002) no. 2-3, pp. 337-362 | MR 1909649 | Zbl 1258.35208 | Zbl 01804060

[16] Lidskiĭ, V. B. Perturbation theory of non-conjugate operators, U.S.S.R. Comput. Math. and Math. Phys., Tome 6 (1966), pp. 73-85 | Article | Zbl 0166.40501

[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 2003421 | Zbl 1061.35186

[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., Tome 18 (1997) no. 4, pp. 793-817 | Article | MR 1471994 | Zbl 0889.15016

[19] Sjöstrand, Johannes Operators of principal type with interior boundary conditions, Acta Math., Tome 130 (1973), pp. 1-51 | Article | MR 436226 | Zbl 0253.35076

[20] Sjöstrand, Johannes Pseudospectrum for differential operators, Seminaire: Équations aux Dérivées Partielles, 2002–2003, École Polytech., Palaiseau (Sémin. Équ. Dériv. Partielles) (2003), pp. Exp. No. XVI, 9 | Numdam | MR 2030711 | Zbl 1061.35067

[21] Sjöstrand, Johannes; Vodev, Georgi Asymptotics of the number of Rayleigh resonances, Math. Ann., Tome 309 (1997) no. 2, pp. 287-306 (With an appendix by Jean Lannes) | Article | MR 1474193 | Zbl 0890.35098

[22] Sjöstrand, Johannes; Zworski, Maciej Asymptotic distribution of resonances for convex obstacles, Acta Math., Tome 183 (1999) no. 2, pp. 191-253 | Article | MR 1738044 | Zbl 0989.35099

[23] Sjöstrand, Johannes; Zworski, Maciej Quantum monodromy and semi-classical trace formulae, J. Math. Pures Appl. (9), Tome 81 (2002) no. 1, pp. 1-33 (See also Quantum monodromy revisited, math.berkeley.edu/~zworski/qmr.ps) | MR 1994881 | Zbl 1038.58033

[24] Trefethen, Lloyd N. Pseudospectra of linear operators, SIAM Rev., Tome 39 (1997) no. 3, pp. 383-406 | Article | MR 1469941 | Zbl 0896.15006

[25] Zelditch, S. Survey on the inverse spectral problem (to appear) | Zbl 1061.58029

[26] Zworski, Maciej Resonances in physics and geometry, Notices of the AMS, Tome 46 (1999) no. 3, pp. 319-328 | MR 1668841 | Zbl 1177.58021

[27] Zworski, Maciej Numerical linear algebra and solvability of partial differential equations, Comm. Math. Phys., Tome 229 (2002) no. 2, pp. 293-307 | Article | MR 1923176 | Zbl 1021.35077