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Sharp Semiclassical Bounds for the Moments of Eigenvalues forSome Schrödinger Type Operators with Unbounded Potentials

Published online by Cambridge University Press:  28 January 2013

V. Vougalter
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Abstract

We establish sharp semiclassical upper bounds for the moments of some negative powers forthe eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporatedin the problem, we obtain sharp lower bounds for the moments of positive powers notexceeding one for such eigenvalues. When considering a Schrödinger operator with therelativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove thesharp Lieb-Thirring estimate for the moments of some negative powers of itseigenvalues.

Keywords

semiclassical boundsLieb-Thirring inequalitiesunbounded potentialsmagnetic fields

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Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 237 - 245
Copyright
© EDP Sciences, 2013

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References

Références

Berezin, F.. Covariant and contravariant symbols of operators. Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134–1167. Google Scholar
Daubechies, I.. An uncertainty principle for fermions with generalized kinetic energy. Comm. Math. Phys., 90 (1983), No. 4, 511–520. CrossRefGoogle Scholar
Dolbeault, J., Felmer, P., Loss, M., Paturel, E.. Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems. J. Funct. Anal., 238 (2006), No. 1, 193–220., CrossRefGoogle Scholar
Erdös, L., Loss, M., Vougalter, V.. Diamagnetic behavior of sums of Dirichlet eigenvalues. Ann. Inst. Fourier, 50 (2000), No. 3, 891–907. CrossRefGoogle Scholar
Frank, R., Loss, M., Weidl, T.. Polya’s conjecture in the presence of a constant magnetic field. J. Eur. Math. Soc. (JEMS), 11 (2009), No. 6, 1365–1383. CrossRefGoogle Scholar
Kovarik, H., Vugalter, S., Weidl, T.. Two-dimensional Berezin-Li-Yau inequalities with a correction term. Comm. Math. Phys., 287 (2009), No. 3, 959–981. CrossRefGoogle Scholar
Li, P., Yau, S.-T.. On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys., 88 (1983), No. 3, 309–318. CrossRefGoogle Scholar
E.H. Lieb, M. Loss. Analysis, Graduate Studies in Mathematics. 14, American Mathematical Society, Providence, (1997)
Lieb, E.H., Loss, M., Siedentop, H.. Stability of relativistic matter via Thomas-Fermi theory. Helv. Phys. Acta, 69 (1996), No. 5–6, 974–984. Google Scholar
A. Laptev, T. Weidl. Recent results on Lieb-Thirring inequalities. Journees “Equations aux Derivees Partielles” (La Chapelle sur Erdre, 2000), Exp.No XX, 14pp., Univ. Nantes, Nantes, (2000)