References
Aizenmann, M. &Lieb, E. H., On semi-classical bounds for eigenvalues of Schrödinger operators.Phys. Lett. A, 66 (1978), 427–429.
Berezin, F. A., Covariant and contravariant symbols of operators.Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 1134–1167 (Russian); English translation inMath. USSR-Izv., 6 (1972), 1117–1151.
—, Conve functions of operators,Mat. Sb. (N.S.), 88 (130) (1972), 268–276.
Blanchard, Ph. &Stubbe, J.,Bound states for Schrödinger Hamiltonians: phase space methods and applications.Rev. Math. Phys., 8 (1996), 503–547.
Bretèche, R. de la, Preuve de la conjecture de Lieb-Thirring dans le cas des potentiels quadratiques strictement convexes.Ann. Inst. H. Poincaré Phys. Théor., 70 (1999), 369–380.
Buslaev, V. S. &Faddeev, L. D., Formulas for traces for a singular Sturm-Liouville differential operator.Dokl. Akad. Nauk SSSR, 132 (1960), 13–16 (Russian); English translation inSoviet Math. Dokl., 1 (1960), 451–454.
Conlon, J. G., A new proof of the Cwikel-Lieb-Rosenbljum bound.Rocky Mountain J. Math., 15 (1985), 117–122.
Cwikel, M., Weak type estimates for singular values and the number of bound states of Schrödinger operators.Ann. of Math., 106 (1977), 93–100.
Faddeev, L. D. &Zakharov, V. E., The Korteweg-de Vries equation is a completely integrable Hamiltonian system.Funktsional. Anal. i Prilozhen., 5 (1971), 18–27 (Russian); English translation inFunctional Anal. Appl., 5 (1971), 280–287.
Glaser, V., Grosse, H. &Martin, A., Bounds on the number of eigenvalues of the Schrödinger operator.Comm. Math. Phys., 59 (1978), 197–212.
Helffer, B., Private communication.
Helffer, B. &Robert, D., Riesz means of bounded states and semi-classical limit connected with a Lieb-Thirring conjecture, I; II.Asymptotic Anal., 3 (1990), 91–103;Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 139–147.
Hundertmark, D., Laptev, A. & Weidl, T., New bounds on the Lieb-Thirring constants. To appear inInvent. Math.
Hundertmark, D., Lieb, E. H. &Thomas, L. E., A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator.Adv. Theor. Math. Phys., 2 (1998), 719–731.
Kato, T., Schrödinger operators with singular potentials.Israel J. Math., 13 (1973), 135–148.
Laptev, A., Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces.J. Funct. Anal., 151 (1997), 531–545.
—, On inequalities for the bound states of Schrödinger operators, inPartial Differential Operators and Mathematical Physics (Holzhau, 1994), pp. 221–225. Oper. Theory Adv. Appl., 78, Birkhäuser, Basel, 1995.
Laptev, A. &Safarov, Yu., A generalization of the Berezin-Lieb inequality, inContemporary Mathematical Physics, pp. 69–79. Amer. Math. Soc. Transl. Ser. 2, 175 Amer. Math. Soc., Providence, RI, 1996.
Li, P. &Yau, S.-T., On the Schrödinger equation and the eigenvalue problem.Comm. Math. Phys., 88 (1983), 309–318.
Lieb, E. H., Bounds on the eigenvalues of the Laplace and Schrödinger operators.Bull. Amer. Math. Soc., 82 (1976), 751–753; See also: The number of bound states of one-body Schrödinger operators and the Weyl problem, inGeometry of the Laplace Operator (Honolulu, HI, 1979), pp. 241–252. Proc. Sympos. Pure Math., 36. Amer. Math. Soc., Providence, RI, 1980.
—, The classical limit of quantum spin systems.Comm. Math. Phys., 31 (1973), 327–340.
—, On characteristic exponents in turbulence.Comm. Math. Phys., 82 (1984), 473–480.
Lieb, E. H. &Thirring, W., Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, inStudies in Mathematical Physics (Essays in Honor of Valentine Bargmann), pp. 269–303. Princeton Univ. Press, Princeton, NJ, 1976.
Pólya, G., On the eigenvalues of vibrating membranes.Proc. London Math. Soc., 11 (1961), 419–433.
Rozenblum, G. V., Distribution of the discrete spectrum of singular differential operators.Dokl. Akad. Nauk SSSR, 202 (1972), 1012–1015 (Russian);Izv. Vyssh. Uchebn. Zaved. Mat., 1976: 1, 75–86 (Russian).
Simon, B., Maximal and minimal Schrödinger forms.J. Operator Theory, 1 (1979), 37–47.
Weidl, T., On the Lieb-Thirring constantsL γ, 1 for\(\gamma \geqslant \tfrac{1}{2}\).Comm. Math. Phys., 178 (1996), 135–146.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Laptev, A., Weidl, T. Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184, 87–111 (2000). https://doi.org/10.1007/BF02392782
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02392782