This text contains a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations. Part III, in collaboration with B. Helffer, gives explicit estimates in the Gearhardt-Prüss theorem for semi-groups.
Ce texte contient une version légèrement completée de mon cours de 6 heures au colloque d’équations aux dérivées partielles à Évian-les-Bains en juin 2009. Dans la première partie on expose quelques résultats anciens et récents sur les opérateurs non-autoadjoints. La deuxième partie est consacrée aux résultats récents sur la distribution de Weyl des valeurs propres des opérateurs elliptiques avec des petites perturbations aléatoires. La partie III, en collaboration avec B. Helffer, donne des bornes explicites dans le théorème de Gearhardt-Prüss pour des semi-groupes.
@incollection{JEDP_2009____A1_0, author = {Sj\"ostrand, Johannes}, title = {Lecture notes : {Spectral} properties of non-self-adjoint operators}, booktitle = {}, series = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {1}, pages = {1--111}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2009}, doi = {10.5802/jedp.54}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jedp.54/} }
TY - JOUR AU - Sjöstrand, Johannes TI - Lecture notes : Spectral properties of non-self-adjoint operators JO - Journées équations aux dérivées partielles PY - 2009 SP - 1 EP - 111 PB - Groupement de recherche 2434 du CNRS UR - http://archive.numdam.org/articles/10.5802/jedp.54/ DO - 10.5802/jedp.54 LA - en ID - JEDP_2009____A1_0 ER -
%0 Journal Article %A Sjöstrand, Johannes %T Lecture notes : Spectral properties of non-self-adjoint operators %J Journées équations aux dérivées partielles %D 2009 %P 1-111 %I Groupement de recherche 2434 du CNRS %U http://archive.numdam.org/articles/10.5802/jedp.54/ %R 10.5802/jedp.54 %G en %F JEDP_2009____A1_0
Sjöstrand, Johannes. Lecture notes : Spectral properties of non-self-adjoint operators. Journées équations aux dérivées partielles (2009), article no. 1, 111 p. doi : 10.5802/jedp.54. http://archive.numdam.org/articles/10.5802/jedp.54/
[1] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math 15(1962), 119–147. | MR | Zbl
[2] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2 D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1965 | MR | Zbl
[3] M.S. Agranovich, A.S. Markus, On spectral properties of elliptic pseudo-differential operators far from selfadjoint ones, Z. Anal. Anwendungen 8 (1989), no. 3, 237–260. | MR | Zbl
[4] M.S. Agranovich, Personal communication, October 2009.
[5] Y. Almog, The stability of the normal state of superconductors in the presence of electric currents, Siam J. Math. Anal. 40 (2)(2008), 824-850. | MR | Zbl
[6] N. Anantharaman, Spectral deviations for the damped wave equation, http://arxiv.org/abs/0904.1736 | MR
[7] V.G. Avakumović, Uber die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z. 65(1956), 324–344. | MR | Zbl
[8] J.M.Bismut, The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc. 18(2005), 379-476. | MR | Zbl
[9] P. Bleher, B. Shiffman, S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142(2)(2000), 351–395. | MR | Zbl
[10] W. Bordeaux Montrieux, Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, thèse, CMLS, Ecole Polytechnique, 2008. http://pastel.paristech.org/5367/
[11] W. Bordeaux Montrieux, J. Sjöstrand, Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds, http://arxiv.org/abs/0903.2937
[12] L.S. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra, J. Operator Theory 47(2)(2002), 413-429. | MR | Zbl
[13] N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2)(2004), 443-471. | MR | Zbl
[14] T. Carleman, Über die asymptotische Verteilung der Eigenwerte partielle Differentialgleichungen, Berichten der mathematisch-physisch Klasse der Sächsischen Akad. der Wissenschaften zu Leipzig, LXXXVIII Band, Sitsung v. 15. Juni 1936. | Zbl
[15] T.W. Cherry, On the solution of Hamiltonian systems of differential equations in the neighboorhood of a singular point, Proc. London. Math. Soc. 27(1928), 151-170.
[16] T. Christiansen, Several complex variables and the distribution of resonances in potential scattering, Comm. Math. Phys. 259(3)(2005), 711–728. | MR | Zbl
[17] T. Christiansen, Several complex variables and the order of growth of the resonance counting function in Euclidean scattering, Int. Math. Res. Not. 2006, Art. ID 43160, 36 pp | MR | Zbl
[18] T. Christiansen, P.D. Hislop, The resonance counting function for Schrödinger operators with generic potentials, Math. Res. Lett. 12(5–6)(2005), 821–826. | MR | Zbl
[19] T.J. Christiansen, M. Zworski, Probabilistic Weyl laws for quantized tori, http://arxiv.org/abs/0909.2014 | MR
[20] Y. Colin de Verdière, Quasi-modes sur les variétés Riemanniennes, Inv. Math. 43(1977), 15–52. | MR | Zbl
[21] E.B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys. 200(1)(1999), 35–41. | MR | Zbl
[22] E.B. Davies, Pseudospectra, the harmonic oscillator and complex resonances, Proc. Roy. Soc. London Ser. A 455(1999), 585–599. | MR | Zbl
[23] E.B. Davies, Linear operators and their spectra, Cambridge Studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge, 2007. | MR | Zbl
[24] E.B. Davies, Semigroup growth bounds, J. Op. Theory 53(2)(2005), 225–249. | MR | Zbl
[25] E.B. Davies, M. Hager, Perturbations of Jordan matrices, J. Approx. Theory 156(1)(2009), 82–94. | MR | Zbl
[26] N. Dencker, J. Sjöstrand, M. Zworski, Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math. 57(3)(2004), 384–415. | MR | Zbl
[27] M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999. | MR | Zbl
[28] L. Desvillettes, C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math., 54(1)(2001), 1–42. | MR | Zbl
[29] J.P. Eckmann, M. Hairer, Spectral properties of hypoelliptic operators, Comm. Math. Phys. 235(2)(2003), 233–253. | MR | Zbl
[30] K.J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. | MR | Zbl
[31] K.J. Engel, R. Nagel, A short course on operator semi-groups, Unitext, Springer-Verlag (2005). | Zbl
[32] I. Gallagher, Th. Gallay, F. Nier, Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator, Int. Math. Res. Not. IMRN 2009, no. 12, 2147–2199. | MR | Zbl
[33] I.C. Gohberg, M.G. Krein, Introduction to the theory of linear non-selfadjoint operators, Translations of mathematical monographs, Vol 18, AMS, Providence, R.I. (1969). | MR | Zbl
[34] S. Graffi, C. Villegas Blas, A uniform quantum version of the Cherry theorem, Comm. Math. Phys. 278(1)(2008), 101–116. | MR | Zbl
[35] A. Grigis, J. Sjöstrand, Microlocal analysis for differential operators, London Math. Soc. Lecture Notes Ser., 196, Cambridge Univ. Press, (1994). | MR | Zbl
[36] M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints, Thesis (2005), http://tel.ccsd.cnrs.fr/docs/00/04/87/08/PDF/tel-00010848.pdf
[37] M. Hager, Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle, Ann. Fac. Sci. Toulouse Math. (6)15(2)(2006), 243–280. | Numdam | MR | Zbl
[38] M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré, 7(6)(2006), 1035–1064. | MR | Zbl
[39] M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen, 342(1)(2008), 177–243. | MR | Zbl
[40] B. Helffer, On spectral problems related to a time dependent model in superconductivity with electric current, Proceedings of the conference in PDE in Evian, June 2009, to appear.
[41] B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics, 1862, Springer-Verlag, Berlin, 2005. | MR | Zbl
[42] B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Comm. Partial Differential Equations 9(4) (1984), 337–408. | MR | Zbl
[43] B. Helffer, J. Sjöstrand, Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Comm. Partial Differential Equations 10(3)(1985), 245–340. | MR | Zbl
[44] B. Helffer, J. Sjöstrand, Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) 24–25(1986), | Numdam | MR | Zbl
[45] F. Hérau, M. Hitrik, J. Sjöstrand, Tunnel effect for Fokker-Planck type operators, Annales Henri Poincaré, 9(2)(2008), 209–274. | MR | Zbl
[46] F. Hérau, M. Hitrik, J. Sjöstrand, Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications, International Math Res Notices, Vol. 2008, Article ID rnn057, 48p. | MR | Zbl
[47] F. Hérau, F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. | MR | Zbl
[48] F. Hérau, J. Sjöstrand, C. Stolk, Semiclassical analysis for the Kramers-Fokker-Planck equation, Comm. PDE 30(5–6)(2005), 689–760. | MR | Zbl
[49] M. Hitrik, Eigenfunctions and expansions for damped wave equations, Meth. Appl. Anal. 10 (4)(2003), 1-22. | MR | Zbl
[50] M. Hitrik, Boundary spectral behavior for semiclassical operators in dimension one, Int. Math. Res. Not. 2004, no. 64, 3417–3438. | MR | Zbl
[51] M. Hitrik, L. Pravda-Starov, Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann. 344(4)(2009), 801–846. | MR | Zbl
[52] M. Hitrik, J. Sjöstrand, Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2, Annales Sci ENS, sér. 4, 41(4)(2008), 511-571. | Numdam | MR | Zbl
[53] M. Hitrik, J. Sjöstrand, S. Vũ Ngọc, Diophantine tori and spectral asymptotics for non-selfadjoint operators, Amer. J. Math. 129(1)(2007), 105–182. | MR | Zbl
[54] L. Hörmander, Differential equations without solutions, Math. Ann. 140(1960), 169–173. | MR | Zbl
[55] L. Hörmander, Differential operators of principal type, Math. Ann. 140(1960), 124–146. | MR | Zbl
[56] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. | MR | Zbl
[57] L. Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, Série des Conférences de l’Union Mathématique Internationale, No. 1. Monographie No. 18 de l’Enseignement Mathématique. Secrétariat de l’Enseignement Mathématique, Université de Genève, Geneva, 1971. 69 pp. | MR | Zbl
[58] L. Hörmander, The analysis of linear partial differential operators. I–IV, Grundlehren der Mathematischen Wissenschaften 256, 257, 274, 275, Springer-Verlag, Berlin, 1983, 1985. | MR | Zbl
[59] D. Jerison, Locating the first nodal line in the Neumann problem, Trans. Amer. Math. Soc. 352(5)(2000), 2301–2317. | MR | Zbl
[60] M.V. Keldysh, On the eigenvalues and eigenfunctions of certain classes of nonselfadjoint equations, Dokl. Akad. Nauk SSSR 77(1951),11–14. English translation in [71] | Zbl
[61] M.V. Keldysh, On a Tauberian theorem, Trudy Mat. Inst. Steklov, 38(1951), 77–86; English transl. in AMS Transl. (2) 102(1973) | MR | Zbl
[62] M.V. Keldysh, On completeness of the eigenfunctions for certain classes of nonselfadjoint linear operators, Uspekhi Mat. Nauk 27(1971)no. 4(160), 15–41. | MR | Zbl
[63] V.V. Kučerenko, Asymptotic solutions of equations with complex characteristics, (Russian) Mat. Sb. (N.S.) 95(137)(1974), 163–213, 327. | MR | Zbl
[64] B. Lascar, J. Sjöstrand, Equation de Schrödinger et propagation des singularités pour des opérateurs pseudodifférentiels à caractéristiques réelles de multiplicité variable I, Astérisque, 95(1982), 467–523. | Numdam | Zbl
[65] V.F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions. With an addendum by A.I. Shnirelman. Ergebnisse der Mathematik und ihrer Grenzgebiete, 24. Springer-Verlag, Berlin, 1993. | MR | Zbl
[66] G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 73–109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. | MR | Zbl
[67] G. Lebeau, Le bismutien, Sém. é.d.p., Ecole Pol. 2004–05,I.1–I.15 | Numdam | MR
[68] B.Ya. Levin, Distribution of zeros of entire functions, English translation, Amer. Math. Soc., Providence, R.I., 1980 | MR | Zbl
[69] B.M. Levitan, Some questions of spectral theory of selfadjoint differential operators, Uspehi Mat. Nauk (N.S.) 11 no. 6 (72)(1956), 117–144. | MR | Zbl
[70] A.J. Lichtenberg, M.A. Lieberman, Regular and chaotic dynamics, Second edition. Springer-Verlag, New York, 1992. | MR | Zbl
[71] A.S. Markus, Introduction to the spectral theory of polynomial operator pencils, Translated from the Russian by H.H. McFaden. Translation edited by Ben Silver. With an appendix by M. V. Keldysh. Translations of Mathematical Monographs, 71. American Mathematical Society, Providence, RI, 1988. | MR | Zbl
[72] A.S. Markus, V.I. Matseev, Asymptotic behavior of the spectrum of close-to-normal operators, Funktsional. Anal. i Prilozhen. 13(3)(1979), 93–94, Functional Anal. Appl. 13(3)(1979), 233–234 (1980). | MR | Zbl
[73] J. Martinet, Sur les propriétés spectrales d’opérateurs nonautoadjoints provenant de la mécanique des fluides, Thèse de doctorat, Université de Paris Sud, 2009.
[74] V.P. Maslov, Operational methods, Translated from the Russian by V. Golo, N. Kulman and G. Voropaeva. Mir Publishers, Moscow, 1976. | MR | Zbl
[75] O. Matte, Correlation asymptotics for non-translation invariant lattice spin systems, Math. Nachr. 281(5)(2008), 721–759. | MR | Zbl
[76] A. Melin, J. Sjoestrand, Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem, Comm. Partial Differential Equations 1(4)(1976), 313–400. | MR | Zbl
[77] A. Melin, J. Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods and Applications of Analysis, 9(2)(2002), 177-238. | MR | Zbl
[78] A. Melin, J. Sjöstrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Astérisque, 284(2003), 181–244. | Numdam | MR | Zbl
[79] A. Menikoff, J. Sjöstrand, On the eigenvalues of a class of hypo-elliptic operators, Math. Ann. 235(1978), 55-85. | MR | Zbl
[80] J. Moser, On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math. 11(1958), 257–271. | MR | Zbl
[81] A. Pazy, Semigroups of linear operators and applications to partial differential operators. Appl. Math. Sci. Vol. 44, Springer (1983). | MR | Zbl
[82] K. Pravda-Starov, Étude du pseudo-spectre d’opérateurs non auto-adjoints, thesis 2006, http://tel.archives-ouvertes.fr/docs/00/10/98/95/PDF/manuscrit.pdf
[83] K. Pravda-Starov, A complete study of the pseudo-spectrum for the rotated harmonic oscillator, J. London Math. Soc. (2) 73(3)(2006), 745–761. | MR | Zbl
[84] K. Pravda-Starov, Boundary pseudospectral behaviour for semiclassical operators in one dimension, Int. Math. Res. Not. IMRN 2007, no. 9, Art. ID rnm 029, 31 pp. | MR | Zbl
[85] K. Pravda-Starov, On the pseudospectrum of elliptic quadratic differential operators, Duke Math. J. 145(2)(2008), 249–279. | MR | Zbl
[86] D. Robert, Autour de l’approximation semi-classique, Progress in Mathematics, 68. Birkhäuser Boston, Inc., Boston, MA, 1987. | MR | Zbl
[87] M. Rudelson, Invertibility of random matrices: norm of the inverse, Ann.of Math. 168(2)(2008), 575–600. | MR | Zbl
[88] E. Schenk, Resonances distribution in partially open quantum chaotic systems, (joint work with Stéphane Nonnenmacher) Talk on November 28th, 2008 at Nice days of waves in complex media.
[89] E. Schenk, Systèmes quantiques ouverts et méthodes semi-classiques, thèse novembre 2009. http://www.lpthe.jussieu.fr/ schenck/thesis.pdf
[90] B. Shiffman, S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200(3)(1999), 661–683. | MR | Zbl
[91] R.T. Seeley, Complex powers of an elliptic operator. 1967 Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288–307 Amer. Math. Soc., Providence, R.I. | MR | Zbl
[92] R. Seeley, A simple example of spectral pathology for differential operators, Comm. Partial Differential Equations 11(6)(1986), 595–598. | MR | Zbl
[93] B. Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. (2) 120(1)(1984), 89–118. | MR | Zbl
[94] J. Sjöstrand, Singularités analytiques microlocales, Astérisque, 95(1982). | Numdam | MR | Zbl
[95] J. Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr., 221(2001), 95–149. | MR | Zbl
[96] J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. of RIMS Kyoto Univ., 36(5)(2000), 573–611. | MR | Zbl
[97] J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations, Ann. Fac. Sci Toulouse, to appear http://arxiv.org/abs/0802.3584 | Numdam | MR | Zbl
[98] J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations, http://arxiv.org/abs/0809.4182 | Numdam | MR
[99] J. Sjöstrand, Resolvent estimates for non-self-adjoint operators via semi-groups, http://arxiv.org/abs/0906.0094 pages 359–384 in International Mathematical Series Vol 13, Around the research of Vladimir Maz’ya III, Springer, Tamara Rozhkovskaya Publisher, 2010 http://arxiv.org/abs/0906.0094 | MR | Zbl
[100] J. Sjöstrand, Counting zeros of holomorphic functions of exponential growth, http://arxiv.org/abs/0910.0346 | MR
[101] J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, Annales Inst. Fourier, 57 (7) (2007), 2095–2141. http://arxiv.org/math.SP/0312166. | Numdam | MR | Zbl
[102] J. Sjöstrand, M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math J, 137 (3) (2007), 381-459. | MR
[103] J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, http://arxiv.org/abs/math/0312166, Ann. Inst. Fourier, to appear. | Numdam | MR | Zbl
[104] J. Tailleur, S. Tanase-Nicola, J. Kurchan, Kramers equation and supersymmetry, J. Stat. Phys. 122(4)(2006), 557–595. | MR | Zbl
[105] L.N. Trefethen, Pseudospectra of linear operators, SIAM Rev. 39(3)(1997), 383–406. | MR | Zbl
[106] L.N. Trefethen, M. Embree, Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, Princeton, NJ, 2005. | MR | Zbl
[107] C. Villani, Hypocoercivity. Memoirs of the AMS, Vol. 202, no. 950 (2009). | MR | Zbl
[108] S. Vũ Ngọc, Systèmes intégrables semi-classiques: du local au global, Panoramas et Synthèses, 22. Société Mathématique de France, Paris, 2006. | MR | Zbl
[109] G. Weiss, The resolvent growth assumption for semigroups on Hilbert spaces, J. Math. An. Appl. 145(1990), 154–171. | MR | Zbl
[110] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71(4)(1912), 441–479 | MR
[111] J. Wunsch, M. Zworski, The FBI transform on compact manifolds, Trans. A.M.S., 353(3)(2001), 1151–1167. | MR | Zbl
[112] M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73(2)(1987), 277–296. | MR | Zbl
[113] M. Zworski, A remark on a paper of E. B Davies: “Semi-classical states for non-self-adjoint Schrödinger operators”, Comm. Math. Phys. 200(1)(1999), 35–41 Proc. Amer. Math. Soc. 129 (10) (2001), 2955–2957 | MR | Zbl
Cited by Sources: