Lower bounds for pseudo-differential operators
Lerner, Nicolas ; Nourrigat, Jean
Annales de l'Institut Fourier, Tome 40 (1990), p. 657-682 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Cet article traite des estimations pour la borne inférieure du spectre d’opérateurs pseudo-différentiels dont les symboles prennent des valeurs négatives. La positivité des pmoyennes du symbole sur des images symplectiques de la boule unité permet d’obtenir une inégalité de type Gårding pour des opérateurs de Schrödinger avec champ magnétique et des opérateurs pseudo-différentiels en dimension un.

This paper contains some new results on lower bounds for pseudo-differential operators whose symbols do not remain positive. Non-negativity of averages of the symbol on canonical images of the unit ball is sufficient to get a Gårding type inequality for Schrödinger operators with magnetic potential and one dimensional pseudo-differential operators.

@article{AIF_1990__40_3_657_0,
     author = {Lerner, Nicolas and Nourrigat, Jean},
     title = {Lower bounds for pseudo-differential operators},
     journal = {Annales de l'Institut Fourier},
     volume = {40},
     year = {1990},
     pages = {657-682},
     doi = {10.5802/aif.1227},
     mrnumber = {92a:35172},
     zbl = {0703.35182},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1990__40_3_657_0}
}

Lerner, Nicolas; Nourrigat, Jean. Lower bounds for pseudo-differential operators. Annales de l'Institut Fourier, Tome 40 (1990) pp. 657-682. doi : 10.5802/aif.1227. http://gdmltest.u-ga.fr/item/AIF_1990__40_3_657_0/

[1] A. Cordoba, C. Fefferman, Wave packets and Fourier integral operators, Comm. PDE, 3, 11 (1978), 979-1005. | MR 80a:35117 | Zbl 0389.35046

[2] C.L. Fefferman, The uncertainty principle, Bull. AMS, 9 (1983), 129-206. | MR 85f:35001 | Zbl 0526.35080

[3] C. Fefferman, D.H. Phong, On positivity of pseudo-differential operators, Proc. Natl. Ac. Sc. USA, 75 (1978), 4673-4674. | MR 80b:47064 | Zbl 0391.35062

[4] C. Fefferman, D.H. Phong, On the lowest eigenvalue of a pseudo-differential operator, Proc. Natl. Ac. Sc. USA, 76 (1979), 6055-6056. | MR 81d:47032 | Zbl 0434.35071

[5] C. Fefferman, D.H. Phong, On the asymptotic eigenvalue distribution, Proc. Natl. Ac. Sc. USA, 77 (1980), 5622-5625. | MR 82h:35101 | Zbl 0443.35082

[6] C. Fefferman, D.H. Phong, The uncertainty principle and sharp Garding inequalities, CPAM, 34 (1981), 285-331. | MR 82j:35140 | Zbl 0458.35099

[7] C. Fefferman, D.H. Phong, Symplectic geometry and positivity of pseudo-differential operators, Proc. Natl. Ac. Sc. USA, 79 (1982), 710-713. | MR 83f:35108 | Zbl 0553.35089

[8] B. Helffer, J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Math. 58, Birkhauser, 1985. | MR 88i:35029 | Zbl 0568.35003

[9] L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math., 83 (1966), 129-209. | MR 38 #1387 | Zbl 0132.07402

[10] L. Hörmander, The Weyl calculus of pseudo-differential operators, CPAM, 32 (1979), 359-443. | Zbl 0388.47032

[11] L. Hörmander, The analysis of linear partial differential operators, four volumes, Berlin, Springer, 1985. | Zbl 0601.35001

[12] P.D. Lax, L. Nirenberg, On stability for difference schemes : a sharp form of Garding's inequality, CPAM 19 (1966), 473-492. | MR 34 #6352 | Zbl 0185.22801

[13] A. Mohamed, J. Nourrigat, Encadrement du N(λ) pour un opérateur de Schrödinger avec des champs électromagnétiques, to appear J. Math. Pures Appl. | Zbl 0725.35068

[14] J. Nourrigat, Subelliptic systems, to appear in Comm. PDE. | Zbl 0723.35089