Lower bounds for pseudo-differential operators
Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 657-682.

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.

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.

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Lerner, Nicolas; Nourrigat, Jean. Lower bounds for pseudo-differential operators. Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 657-682. doi : 10.5802/aif.1227. https://aif.centre-mersenne.org/articles/10.5802/aif.1227/

[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

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