Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.
Annales Mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 197-211.

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.

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.
@article{AMBP_2012__19_1_197_0,
     author = {Mohammed El A\"\i di},
     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},
     pages = {197--211},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {19},
     number = {1},
     year = {2012},
     doi = {10.5802/ambp.310},
     mrnumber = {2978319},
     zbl = {1256.35034},
     language = {fr},
     url = {ambp.centre-mersenne.org/item/AMBP_2012__19_1_197_0/}
}
Mohammed El Aïdi. Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.. Annales Mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 197-211. doi : 10.5802/ambp.310. https://ambp.centre-mersenne.org/item/AMBP_2012__19_1_197_0/

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

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

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

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

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

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

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

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

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

[11] Y.V. Egorov; V.A. Kondratiev 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, Volume 150 (1992), pp. 129-206 | MR 1157650 | Zbl 0756.35058

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

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

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

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

[17] B. Helffer; J. Sjöstrand Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) (1986) no. 24-25, iv+228 pages | Numdam | MR 871788 | Zbl 0631.35075

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

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

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

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

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

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

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

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

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

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

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

[29] M. Skriganov 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), Volume 113 (1980), pp. 131-145 | MR 590542 | Zbl 0464.35064

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

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

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

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