Diamagnetic behavior of sums Dirichlet eigenvalues
Erdös, László ; Loss, Michael ; Vougalter, Vitali
Annales de l'Institut Fourier, Tome 50 (2000), p. 891-907 / Harvested from Numdam
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Nous étendons la borne inférieure semi-classique due à Li-Yau pour la somme des N premières valeurs propres du laplacien de Dirichlet aux laplaciens de Dirichlet avec un champ magnétique constant. Notre méthode repose sur une nouvelle inégalité pour les champs magnétiques constants.

The Li-Yau semiclassical lower bound for the sum of the first N eigenvalues of the Dirichlet–Laplacian is extended to Dirichlet– Laplacians with constant magnetic fields. Our method involves a new diamagnetic inequality for constant magnetic fields.

@article{AIF_2000__50_3_891_0,
     author = {Erd\"os, L\'aszl\'o and Loss, Michael and Vougalter, Vitali},
     title = {Diamagnetic behavior of sums Dirichlet eigenvalues},
     journal = {Annales de l'Institut Fourier},
     volume = {50},
     year = {2000},
     pages = {891-907},
     doi = {10.5802/aif.1777},
     mrnumber = {2001g:35201},
     zbl = {0957.35104},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2000__50_3_891_0}
}

Erdös, László; Loss, Michael; Vougalter, Vitali. Diamagnetic behavior of sums Dirichlet eigenvalues. Annales de l'Institut Fourier, Tome 50 (2000) pp. 891-907. doi : 10.5802/aif.1777. http://gdmltest.u-ga.fr/item/AIF_2000__50_3_891_0/

[AHS78] J.E. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields, I. General interactions, Duke Math. J., 45 (1978), 847-883. | MR 80k:35054 | Zbl 0399.35029

[B72] F.A. Berezin, Convex operator functions, Math. USSR. Sb., 17 (1972), 269-277. | Zbl 0279.47006

[CFKS87] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrödinger operators (with application to Quantum Mechanics and Global Geometry), Springer 1987. | Zbl 0619.47005

[HSU77] H. Hess, R. Schrader and D.A. Uhlenbrock, Domination of semigroups and generalizations of Kato's inequality, Duke Math. J., 44 (1977), 893-904. | MR 56 #16446 | Zbl 0379.47028

[Iv98] V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer, 1998. | MR 99e:58193 | Zbl 0906.35003

[K72] T. Kato, Schrödinger operators with singular potentials, Isr. J. Math., 13 (1972), 135-148. | MR 48 #12155 | Zbl 0246.35025

[L80] E.H. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, Proc. Sym. Pure Math., 36 (1980), 241-252. | MR 82i:35134 | Zbl 0445.58029

[LL97] E.H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, Volume 14, American Mathematical Society (1997). | MR 98b:00004 | Zbl 0873.26002

[LT75] E.H. Lieb, W. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett., 35 (1975), 687.

[LT76] E.H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Math. Phys., Essays in Honor of Valentine Bargmann., Princeton (1976). | Zbl 0342.35044

[LW99] A. Laptev, T. Weidl, Sharp Lieb-Thirring inequalities in high dimensions, to appear in Acta Math. | Zbl 01541221

[LY83] P. Li, S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. | MR 84k:58225 | Zbl 0554.35029

[RS79] M. Reed, B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, 1979. | MR 80m:81085 | Zbl 0405.47007

[S79] B. Simon, Functional integration and Quantum Physics, Academic Press, 1979. | MR 84m:81066 | Zbl 0434.28013

[S77, 79] B. Simon, An abstract Kato inequality for generators of positivity preserving semigroups, Ind. Math. J., 26 (1977), 1067-1073. Kato's inequality and the comparison of semigroups, J. Funct. Anal., 32 (1979), 97-101. | MR 57 #1194 | Zbl 0389.47021