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Johannes Sjöstrand; Maciej Zworski
Elementary linear algebra for advanced spectral problems
(Algèbre linéaire élémentaire pour des problèmes d’analyse spectrale)
Annales de l'institut Fourier, 57 no. 7 (2007), p. 2095-2141, doi: 10.5802/aif.2328
Article PDF | Reviews MR 2394537 | Zbl 1140.15009 | 5 citations in Cedram
Class. Math.: 15A21, 35P05, 35Q40, 81Q15
Keywords: Grushin problem, Schur complement, Feshbach reduction, eigenvalues, resonances, trace formulæ
[1] David Bau & Lloyd N. Trefethen, Numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), 1997 MR 1444820 | Zbl 0874.65013
[2] Louis Boutet de Monvel, “Boundary problems for pseudo-differential operators”, Acta Math. 126 (1971) no. 1-2, p. 11-51
Article | MR 407904 | Zbl 0206.39401
[3] E.B. Davies & M. Hager, “Perturbations of Jordan matrices”, arXiv:math/0612158v
arXiv
[4] Jan Dereziński & Vojkan Jakšić, “Spectral theory of Pauli-Fierz operators”, J. Funct. Anal. 180 (2001) no. 2, p. 243-327
Article | MR 1814991 | Zbl 1034.81016
[5] Mouez Dimassi & Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999 MR 1735654 | Zbl 0926.35002
[6] Roger Fletcher & Tom Johnson, “On the stability of null-space methods for KKT systems”, SIAM J. Matrix Anal. Appl. 18 (1997) no. 4, p. 938-958
Article | MR 1472003 | Zbl 0890.65060
[7] I. C. Gohberg & M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, 1969 MR 246142 | Zbl 0181.13504
[8] V. V. Grushin, Les problèmes aux limites dégénérés et les opérateurs pseudo-différentiels, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, 1971, p. 737–743 MR 509187 | Zbl 0244.35075
[9] M. Hager & J. Sjöstrand, “Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators”, arXiv: math.SP/0601381
arXiv
[10] B. Helffer & J. Sjöstrand, “Résonances en limite semi-classique”, Mém. Soc. Math. France (N.S.) (1986) no. 24-25
Numdam | MR 871788 | Zbl 0631.35075
[11] B. Helffer & J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988), Springer, 1989, p. 118–197 MR 1037319 | Zbl 0699.35189
[12] B. Helffer & J. Sjöstrand, “Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum”, Mém. Soc. Math. France (N.S.) (1989) no. 39, p. 1-124
Numdam | Zbl 0725.34099
[13] Lars Hörmander, The analysis of linear partial differential operators. I, II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 256, 257, Springer-Verlag, 1983, Distribution theory and Fourier analysis MR 717035 | Zbl 0521.35001
[14] Lars Hörmander, The analysis of linear partial differential operators. III, IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 274,275, Springer-Verlag, 1985 MR 781536 | Zbl 0601.35001
[15] A. Iantchenko, J. Sjöstrand & M. Zworski, “Birkhoff normal forms in semi-classical inverse problems”, Math. Res. Lett. 9 (2002) no. 2-3, p. 337-362 MR 1909649 | Zbl 01804060
[16] V. B. Lidskiĭ, “Perturbation theory of non-conjugate operators”, U.S.S.R. Comput. Math. and Math. Phys. 6 (1966), p. 73-85
Article | Zbl 0166.40501
[17] Anders Melin & Johannes Sjöstrand, “Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2”, Astérisque (2003) no. 284, p. 181-244, Autour de l’analyse microlocale MR 2003421 | Zbl 1061.35186
[18] Julio Moro, James V. Burke & Michael L. Overton, “On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure”, SIAM J. Matrix Anal. Appl. 18 (1997) no. 4, p. 793-817
Article | MR 1471994 | Zbl 0889.15016
[19] Johannes Sjöstrand, “Operators of principal type with interior boundary conditions”, Acta Math. 130 (1973), p. 1-51
Article | MR 436226 | Zbl 0253.35076
[20] Johannes Sjöstrand, Pseudospectrum for differential operators, Seminaire: Équations aux Dérivées Partielles, 2002–2003, École Polytech., 2003, p. Exp. No. XVI, 9
Cedram | MR 2030711 | Zbl 1061.35067
[21] Johannes Sjöstrand & Georgi Vodev, “Asymptotics of the number of Rayleigh resonances”, Math. Ann. 309 (1997) no. 2, p. 287-306, With an appendix by Jean Lannes
Article | MR 1474193 | Zbl 0890.35098
[22] Johannes Sjöstrand & Maciej Zworski, “Asymptotic distribution of resonances for convex obstacles”, Acta Math. 183 (1999) no. 2, p. 191-253
Article | MR 1738044 | Zbl 0989.35099
[23] Johannes Sjöstrand & Maciej Zworski, “Quantum monodromy and semi-classical trace formulae”, J. Math. Pures Appl. (9) 81 (2002) no. 1, p. 1-33, See also Quantum monodromy revisited,
[24] Lloyd N. Trefethen, “Pseudospectra of linear operators”, SIAM Rev. 39 (1997) no. 3, p. 383-406
Article | MR 1469941 | Zbl 0896.15006
[25] S. Zelditch, “Survey on the inverse spectral problem”, to appear Zbl 1061.58029
[26] Maciej Zworski, “Resonances in physics and geometry”, Notices of the AMS 46 (1999) no. 3, p. 319-328 MR 1668841 | Zbl 1177.58021
[27] Maciej Zworski, “Numerical linear algebra and solvability of partial differential equations”, Comm. Math. Phys. 229 (2002) no. 2, p. 293-307
Article | MR 1923176 | Zbl 1021.35077
Johannes Sjöstrand; Maciej Zworski
Elementary linear algebra for advanced spectral problems
(Algèbre linéaire élémentaire pour des problèmes d’analyse spectrale)
Annales de l'institut Fourier, 57 no. 7 (2007), p. 2095-2141, doi: 10.5802/aif.2328
Article PDF | Reviews MR 2394537 | Zbl 1140.15009 | 5 citations in Cedram
Class. Math.: 15A21, 35P05, 35Q40, 81Q15
Keywords: Grushin problem, Schur complement, Feshbach reduction, eigenvalues, resonances, trace formulæ
Résumé - Abstract
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.
Bibliography
[2] Louis Boutet de Monvel, “Boundary problems for pseudo-differential operators”, Acta Math. 126 (1971) no. 1-2, p. 11-51
Article | MR 407904 | Zbl 0206.39401
[3] E.B. Davies & M. Hager, “Perturbations of Jordan matrices”, arXiv:math/0612158v
arXiv
[4] Jan Dereziński & Vojkan Jakšić, “Spectral theory of Pauli-Fierz operators”, J. Funct. Anal. 180 (2001) no. 2, p. 243-327
Article | MR 1814991 | Zbl 1034.81016
[5] Mouez Dimassi & Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999 MR 1735654 | Zbl 0926.35002
[6] Roger Fletcher & Tom Johnson, “On the stability of null-space methods for KKT systems”, SIAM J. Matrix Anal. Appl. 18 (1997) no. 4, p. 938-958
Article | MR 1472003 | Zbl 0890.65060
[7] I. C. Gohberg & M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, 1969 MR 246142 | Zbl 0181.13504
[8] V. V. Grushin, Les problèmes aux limites dégénérés et les opérateurs pseudo-différentiels, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, 1971, p. 737–743 MR 509187 | Zbl 0244.35075
[9] M. Hager & J. Sjöstrand, “Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators”, arXiv: math.SP/0601381
arXiv
[10] B. Helffer & J. Sjöstrand, “Résonances en limite semi-classique”, Mém. Soc. Math. France (N.S.) (1986) no. 24-25
Numdam | MR 871788 | Zbl 0631.35075
[11] B. Helffer & J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators (Sønderborg, 1988), Springer, 1989, p. 118–197 MR 1037319 | Zbl 0699.35189
[12] B. Helffer & J. Sjöstrand, “Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum”, Mém. Soc. Math. France (N.S.) (1989) no. 39, p. 1-124
Numdam | Zbl 0725.34099
[13] Lars Hörmander, The analysis of linear partial differential operators. I, II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 256, 257, Springer-Verlag, 1983, Distribution theory and Fourier analysis MR 717035 | Zbl 0521.35001
[14] Lars Hörmander, The analysis of linear partial differential operators. III, IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 274,275, Springer-Verlag, 1985 MR 781536 | Zbl 0601.35001
[15] A. Iantchenko, J. Sjöstrand & M. Zworski, “Birkhoff normal forms in semi-classical inverse problems”, Math. Res. Lett. 9 (2002) no. 2-3, p. 337-362 MR 1909649 | Zbl 01804060
[16] V. B. Lidskiĭ, “Perturbation theory of non-conjugate operators”, U.S.S.R. Comput. Math. and Math. Phys. 6 (1966), p. 73-85
Article | Zbl 0166.40501
[17] Anders Melin & Johannes Sjöstrand, “Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2”, Astérisque (2003) no. 284, p. 181-244, Autour de l’analyse microlocale MR 2003421 | Zbl 1061.35186
[18] Julio Moro, James V. Burke & Michael L. Overton, “On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure”, SIAM J. Matrix Anal. Appl. 18 (1997) no. 4, p. 793-817
Article | MR 1471994 | Zbl 0889.15016
[19] Johannes Sjöstrand, “Operators of principal type with interior boundary conditions”, Acta Math. 130 (1973), p. 1-51
Article | MR 436226 | Zbl 0253.35076
[20] Johannes Sjöstrand, Pseudospectrum for differential operators, Seminaire: Équations aux Dérivées Partielles, 2002–2003, École Polytech., 2003, p. Exp. No. XVI, 9
Cedram | MR 2030711 | Zbl 1061.35067
[21] Johannes Sjöstrand & Georgi Vodev, “Asymptotics of the number of Rayleigh resonances”, Math. Ann. 309 (1997) no. 2, p. 287-306, With an appendix by Jean Lannes
Article | MR 1474193 | Zbl 0890.35098
[22] Johannes Sjöstrand & Maciej Zworski, “Asymptotic distribution of resonances for convex obstacles”, Acta Math. 183 (1999) no. 2, p. 191-253
Article | MR 1738044 | Zbl 0989.35099
[23] Johannes Sjöstrand & Maciej Zworski, “Quantum monodromy and semi-classical trace formulae”, J. Math. Pures Appl. (9) 81 (2002) no. 1, p. 1-33, See also Quantum monodromy revisited,
math.berkeley.edu/MR 1994881 | Zbl 1038.58033[24] Lloyd N. Trefethen, “Pseudospectra of linear operators”, SIAM Rev. 39 (1997) no. 3, p. 383-406
Article | MR 1469941 | Zbl 0896.15006
[25] S. Zelditch, “Survey on the inverse spectral problem”, to appear Zbl 1061.58029
[26] Maciej Zworski, “Resonances in physics and geometry”, Notices of the AMS 46 (1999) no. 3, p. 319-328 MR 1668841 | Zbl 1177.58021
[27] Maciej Zworski, “Numerical linear algebra and solvability of partial differential equations”, Comm. Math. Phys. 229 (2002) no. 2, p. 293-307
Article | MR 1923176 | Zbl 1021.35077
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310