Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.
El Aïdi, Mohammed
Annales mathématiques Blaise Pascal, Tome 19 (2012), p. 197-211 / Harvested from Numdam
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On donne une borne supérieur du nombre des valeurs propres négatives de l’opérateur de Schrödinger généralisé, cette borne est donnée en fonction d’un nombre fini de cube dyadiques minimaux.

This paper is devoted to give an upper bound of the number of negative eigenvalues of the generalized Schrödinger operator, and this upper bound is given in terms of a finite number of minimal dyadic cubes.

Publié le : 2012-01-01
DOI : https://doi.org/10.5802/ambp.310
Classification:  34B09,  34L15,  34L25,  34L05,  35J40,  35P15,  35R06,  35R15,  47A75,  47A07,  47A40,  47A10,  57R40,  58D10
Mots clés: Valeurs propres négatives, Principe de minmax. Cubes dyadiques. Potentiel de Riesz. Résonances. 
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     author = {El A\"\i di, Mohammed},
     title = {Un majorant du nombre des valeurs propres n\'egatives correspondantes \`a l'op\'erateur de Schr\"odinger g\'en\'eralis\'e.},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {19},
     year = {2012},
     pages = {197-211},
     doi = {10.5802/ambp.310},
     zbl = {1256.35034},
     mrnumber = {2978319},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AMBP_2012__19_1_197_0}
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El Aïdi, Mohammed. Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.. Annales mathématiques Blaise Pascal, Tome 19 (2012) pp. 197-211. doi : 10.5802/ambp.310. http://gdmltest.u-ga.fr/item/AMBP_2012__19_1_197_0/

[1] Adams, R.A. Sobolev space, Academics Press (1975) | Zbl 0314.46030

[2] Avron, J.; Bender-Wu Formulas for the Zeeman effect in hydrogen, Ann. Phys. Publ. Mat., Tome 131 (1981), pp. 73-94 | Article | MR 608087

[3] Barreto, A. Sá; Zworski, M. Existence of resonances in potential scattering, Commun. Pure Appl. Math., Tome 49 (1996), pp. 1271-1280 | Article | MR 1414586 | Zbl 0877.35087

[4] Bony, Jean-François; Sjöstrand, Johannes Traceformula for resonances in small domains, J. Funct. Anal., Tome 184 (2001) no. 2, pp. 402-418 | Article | MR 1851003

[5] Bony, J.F. Minoration du nombre de résonances engendrées par une trajectoire fermée, Commun. Partial Differ. Equations, Tome 27 No.5-6 (2002), pp. 1021-1078 | Article | MR 1916556

[6] Burq, N. Lower bounds for shape resonances widths of long rang Schrödinger operators, Am. J..Math., Tome 124, No.4 (2002), pp. 677-735 | Article | MR 1914456

[7] Combes, J.M.; Duclos, P.; Klein, M.; Seiler, R. The shape resonance, Comm. Math. Phy., Tome 110 (1987), pp. 215-236 | Article | MR 887996 | Zbl 0629.47044

[8] Dahlberg, B.E.J.; Trubowitz, E. A remark on two dimensional periodic potentials, Comment. Math. Helvetici, Tome 57 (1982), pp. 130-134 | Article | MR 672849 | Zbl 0539.35059

[9] Dolbeault, J.; Flores, I. GEOMETRY OF PHASE SPACE AND SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS IN A BALL, Trans. Amer. Math. Soc., Tome 359 (2007), pp. 4073-4087 | Article | MR 2309176

[10] Egorov, Yu. V.; El Aïdi, M. Spectre négatif d’un opérateur elliptique avec des conditions au bord de Robin, Publ. Mat., Tome 45 (2001) no. 1, pp. 125-148 | Article | MR 1829580

[11] Egorov, Y.V.; Kondratiev, V.A. Estimates of the negative spectrum of an elliptic operator, in Spectral theory of operators, (Novgorod, 1989), Amer.Math.Soc.Transl.Ser.2, Amer.Math. Soc., Providence, RI, Tome 150 (1992), pp. 129-206 | MR 1157650 | Zbl 0756.35058

[12] Fabes, E.; Kenig, C.; Serapioni, R. The local regularity of solutions of degenerate elliptic equations, Comm. in P.D.E., Tome 7 (1982), pp. 77-116 | Article | MR 643158 | Zbl 0498.35042

[13] Fefferman, C.L. The Uncertainty Principle, Bull. A.M.S (1983), pp. 129-206 | Article | MR 707957 | Zbl 0526.35080

[14] Glazman, I.M. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963. English translation : Daniel Davey and Co., New York (1966) | MR 190800

[15] Guo, Z.; Wei, J. Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Amer. Math. Soc., Tome 363 (2011), pp. 4777-4799 | Article | MR 2806691

[16] Harell, E.; Simon, B. The mathematical theory of resonances which have exponentially small widths, Duke Math. J., Tome 47 (1980), pp. 845-902 | MR 596118 | Zbl 0455.35091

[17] 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

[18] Hislop, P. D.; Sigal, I. M. Shape resonances in quantum mechanics, Differential equations and mathematical physics (Birmingham, Ala., 1986), Springer, Berlin (Lecture Notes in Math.) Tome 1285 (1987), pp. 180-196 | Article | MR 921268 | Zbl 0653.46074

[19] Hislop, P.D.; Sigal, I.M. Semiclassical resolvent estimates, Ann. Inst. H.Poincaré Phys. Théor., Tome 51 (1989), pp. 187-198 | Numdam | MR 1033616 | Zbl 0719.35064

[20] Kerman, R.; Sawyer, T. The trace inequality and eigenvalue estimates for Schrödinger operators, Ann.Inst.Fourier,Grenoble(36), Tome 4 (1986), pp. 207-228 | Article | Numdam | MR 867921 | Zbl 0591.47037

[21] Martin, A. Résonance dans l’approximation de Born Oppenheimer I, Journal of Differ. Eq. (1991), pp. 204-234 | Article | MR 1111174 | Zbl 0737.35046

[22] Martin, A. Résonance dans l’approximation de Born Oppenheimer II, Commun.Math.Phys., Tome 135 (1991), pp. 517-530 | Article | MR 1091576 | Zbl 0737.35047

[23] Maz’Ya, Vladimir Sobolev spaces with applications to elliptic partial differential equations, Springer, Heidelberg, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 342 (2011) | Article | MR 2777530

[24] Parnovski, L.; Sobolev, A. On the Beth-Sommerfeld conjecture for the polyharmonic operator, Duke Math. J., Tome 107 Number 2 (2001), pp. 209-238 | MR 1823047

[25] Petkov, V.; Zworski, M. Breit-Wigner approximation and the distribution of resonances, Comm. Math. Phy., Tome 204 (1999), pp. 329-351 (erratum : Comm. Math. Phys. 214 (2000), p. 733-735) | Article | MR 1704278 | Zbl 0936.47004

[26] Popov, V.N.; Skriganov, M. A remark on the spectral structure of the two di- mensional Schrödinger operator with a periodic potential, Zap. Nauchn. Sem. LOMI AN SSSR, Tome 109 (1981), pp. 131-133 | MR 629118 | Zbl 0492.47024

[27] Reed, Michael; Simon, Barry Methods of modern mathematical physics. I, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980) (Functional analysis) | MR 751959 | Zbl 0459.46001

[28] Shubin, M.A. Pseudodifferential Operators and Spectral Theory, Second Edition, Springer-Verlag (2001) | MR 1852334 | Zbl 0616.47040

[29] Skriganov, M. Finiteness of the number of gaps in the spectrum of the mutlidimensional polyharmonic operator with a periodic potential., Mat. Sb (Engl. transl. : Math. USSR Sb. 41 (1982), Tome 113 (1980), pp. 131-145 | MR 590542 | Zbl 0464.35064

[30] Skriganov, M. M. Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov., Tome 171 (1985), pp. 122 | MR 798454 | Zbl 0567.47004

[31] Veliev., O.A. Asymptotic formulas for the eigenvalues of a periodic Schrödinger operator and the Bethe-Sommerfeld conjecture, Functional Anal. Appl., Tome 21 (1987), pp. 87-99 | Article | MR 902289 | Zbl 0638.47049

[32] Verbitsky, I. Nonlinear potentials and trace inequalities, The Maz’ya anniversary collection, Vol2 (Rostock, 1998), 323-343, Oper.Theory Adv.Appl., Birkhäuser, Basel,, Tome 110 (1998), pp. 323-343 | MR 1747901 | Zbl 0941.31001

[33] Zworski, M. Resonances in physics in geometry, Notices Amer. Math. Soc., Tome 46 (1999), pp. 319-328 | MR 1668841 | Zbl 1177.58021