no. 1
Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.
El Aïdi, Mohammed1
1 Departamento de Matemáticas Universidad Nacional de Colombia. Avenida Carrera 30, numéro 45-03. Bogotá, D.C. Colombia.
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 : 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.
El Aïdi, Mohammed 1

1 Departamento de Matemáticas Universidad Nacional de Colombia. Avenida Carrera 30, numéro 45-03. Bogotá, D.C. Colombia. 
@article{AMBP_2012__19_1_197_0,
     author = {El A{\"\i}di, Mohammed},
     title = {Un majorant du nombre des valeurs propres n\'egatives correspondantes \`a l{\textquoteright}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},
     zbl = {1256.35034},
     mrnumber = {2978319},
     language = {fr},
     url = {http://www.numdam.org/articles/10.5802/ambp.310/}
}
TY  - JOUR
AU  - El Aïdi, Mohammed
TI  - Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.
JO  - Annales mathématiques Blaise Pascal
PY  - 2012
SP  - 197
EP  - 211
VL  - 19
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.310/
DO  - 10.5802/ambp.310
LA  - fr
ID  - AMBP_2012__19_1_197_0
ER  - 
%0 Journal Article
%A El Aïdi, Mohammed
%T Un majorant du nombre des valeurs propres négatives correspondantes à l’opérateur de Schrödinger généralisé.
%J Annales mathématiques Blaise Pascal
%D 2012
%P 197-211
%V 19
%N 1
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.310/
%R 10.5802/ambp.310
%G fr
%F AMBP_2012__19_1_197_0
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) no. 1, pp. 197-211. doi : 10.5802/ambp.310. http://www.numdam.org/articles/10.5802/ambp.310/

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

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

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

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

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

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

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

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

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

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

[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, Volume 150 (1992), pp. 129-206 | MR | Zbl

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

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

[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

[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., Volume 363 (2011), pp. 4777-4799 | DOI | MR

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

[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 | Zbl

[18] Hislop, P. D.; Sigal, I. M. 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 | DOI | MR | Zbl

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

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

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

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

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

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

[25] Petkov, V.; Zworski, M. 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 | DOI | MR | Zbl

[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, Volume 109 (1981), pp. 131-133 | MR | Zbl

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

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

[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), Volume 113 (1980), pp. 131-145 | MR | Zbl

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

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

[32] Verbitsky, I. 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 | Zbl

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

Cité par Sources :