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Decay at infinity of eigenfunctions of Schrödinger operators with polynomial electro-magnetic field. (Décroissance à l’infini des fonctions propres de l’opérateur de Schrödinger avec champ électromagnétique polynomial.) (French) Zbl 0814.35080
This paper concerns the exponential decay at infinity of the eigenfunctions of Schrödinger operators with magnetic field. The electromagnetic potentials are assumed to be polynomials, and the decay is expressed in terms of some “Agmon distance” associated to a positive function involving all the derivatives of both electric and magnetic potentials. A semiclassical version of the result is also given.

MSC:
35P05 General topics in linear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35J10 Schrödinger operator, Schrödinger equation
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[1] [A] S. Agmon,Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations, Math. Notes, No. 29, Princeton University Press, 1982. · Zbl 0503.35001
[2] [Av-He-Si] J. Avron, I. Herbst and B. Simon,Schrödinger operators with magnetic fields, I General interactions, Duke Math. J.45 (1978), 847–884. · Zbl 0399.35029 · doi:10.1215/S0012-7094-78-04540-4
[3] [B1] R. Brummelhuis,Exponential decay in the semiclassical limit for eigenfunctions of Schrödinger operators: case of potentials which degenerate at infinity, preprint, July 1990.
[4] [B2] R. Brummelhuis,Exponential decay in the semiclassical limit for eigenfunctions of Schrödinger operators with magnectic fuields and potentials which degenerate at infinity, preprint, October 1990.
[5] [Fe] C. Fefferman,The uncertainty principle, Bull. Am. Math. Soc.9 (1983), 129–206. · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6
[6] [Fe-Ph] C. Fefferman and D. Phong,The uncertainty principle and sharp Garding inequality, Comm. Pures Appl. Math.34 (1981), 285–331. · Zbl 0458.35099 · doi:10.1002/cpa.3160340302
[7] [Gu] D. Guibourg,Une inégalité LB 2 sur l’opératuer de Schrödinger avec des champs électrique et magnétique polynomiaux, manuscript.
[8] [He-Mo] B. Helffer and A. Mohamed,Sur le spectre essentiel des opérateurs de Schrödinger avec champ magnétique, Ann. Inst. Fourier38 (1988), fasc. 2, 95–113.
[9] [He-No] B. Helffer and J. Nourrigat,Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Progress in Math.58, Birkhäuser, Boston, 1985. · Zbl 0568.35003
[10] [He-Sj1] B. Helffer and J. Sjöstrand,Multiple wells in the semi-classical limit I, Comm. Partial Differ. Equ.9(4) (1984), 337–408. · Zbl 0546.35053 · doi:10.1080/03605308408820335
[11] [He-Sj2] B. Helffer and J. Sjöstrand,Effet tunnel pour l’équation de Schrödinger avec champ magnétique, Annales de l’ENS de Pise, Vol. XIV,4 (1987), 625–657. · Zbl 0699.35205
[12] [Hö] L. Hörmander,The analysis of linear partial differential operators I, II, III, to appear.
[13] [Mo-No] A. Mohamed and J. Nourrigat,Encadrement du N(\(\lambda\)) pour des opérateurs de Schrödinger avec champ magnétique, J. Math. Pures Appl.70 (1991), 87–99.
[14] [No] J. Nourrigat,Inégalités L 2 et représentations des groupes nilpotents, J. Funct. Anal.74 (1987), 307–327. · Zbl 0644.35026 · doi:10.1016/0022-1236(87)90027-9
[15] [P] A. Persson,Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand.8 (1960), 143–153. · Zbl 0145.14901
[16] [Si] B. Simon,Instantons, double wells and large deviations, Bull. Am. Math. Soc.8 (1983), 323–326. · Zbl 0529.35059 · doi:10.1090/S0273-0979-1983-15104-2
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