Recent results on Lieb-Thirring inequalities
Journées équations aux dérivées partielles (2000), article no. 20, 14 p.

We give a survey of results on the Lieb-Thirring inequalities for the eigenvalue moments of Schrödinger operators. In particular, we discuss the optimal values of the constants therein for higher dimensions. We elaborate on certain generalisations and some open problems as well.

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Laptev, Ari; Weidl, Timo. Recent results on Lieb-Thirring inequalities. Journées équations aux dérivées partielles (2000), article  no. 20, 14 p. http://www.numdam.org/item/JEDP_2000____A20_0/

[1] Aizenman M. and Lieb E.H.: On semi-classical bounds for eigenvalues of Schrödinger operators. Phys. Lett. 66A, 427-429 (1978) | MR

[2] Benguria R and Loss M.: A simple proof of a theorem by Laptev and Weidl. Preprint (1999). | Zbl

[3] Berezin F.A.: Covariant and contravariant symbols of operators. [English Translation] Math. USSR, 6, 1117-1151 (1972) | MR | Zbl

[4] Birman M.S.: The spectrum of singular boundary problems. (Russian) Mat. Sb. (N.S.) 55 (97), 125-174 (1961). (English) Amer. Math. Soc. Transl. 53, 23-80 (1966) | MR | Zbl

[5] Birman M. S. and Laptev A.: The negative discrete spectrum of a two-dimensional Schrödinger operator. Comm. Pure and Appl. Math., XLIX, 967-997 (1996) | MR | Zbl

[6] Blanchard Ph. and Stubbe J.: Bound states for Schrödinger Hamiltonians: Phase Space Methods and Applications. Rev. Math. Phys., 35, 504-547 (1996) | Zbl

[7] Buslaev V.S. and Faddeev L.D: Formulas for traces for a singular Sturm-Liouville differential operator. [English translation], Dokl. AN SSSR, 132, 451-454 (1960) | MR | Zbl

[8] Cwikel M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Trans. AMS, 224, 93-100 (1977) | MR | Zbl

[9] De La Bretèche R.: Preuve de la conjecture de Lieb-Thirring dans le cas des potentiels quadratiques strictement convexes. Ann. Inst. H. Poincaré Phys. théor., 70, 369-380 (1999) | Numdam | MR | Zbl

[10] Egorov Yu. V. and Kontrat'Ev V.A.: On spectral theory of elliptic operators. Operator Theory: Advances and Applications, 89, Birkhäuser Verlag, Basel, 1996. x+328 pp. | MR | Zbl

[11] Faddeev L.D. and Zakharov V.E.: Korteweg-de Vries equation: A completely integrable hamiltonian system. Func. Anal. Appl., 5, 18-27 (1971) | Zbl

[12] Glaser V., Grosse H. and Martin A.: Bounds on the number of eigenvalues of the Schrödinger operator. Commun. Math. Phys., 59, 197-212 (1978) | MR | Zbl

[13] Helffer B. and Robert D.: Riesz means of bounded states and semi-classical limit connected with a Lieb-Thirring conjecture I, II. I -Jour. Asymp. Anal., 3, 91-103 (1990), II - Ann. de l'Inst. H. Poincare, 53 (2), 139-147 (1990) | Numdam | MR | Zbl

[14] Hundertmark D., Laptev A. and Weidl T.: New bounds on the Lieb-Thirring constants. to appear in Inventiones mathematicae. | Zbl

[15] Hundertmark D., Lieb E.H. and Thomas L.E.: A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. Adv. Theor. Math. Phys. 2, 719-731 (1998) | MR | Zbl

[16] Laptev A.: Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces. J. Func. Anal., 151, 531-545 (1997) | MR | Zbl

[17] Laptev A.: On the Lieb-Thirring conjecture for a class of potentials. Operator Theory: Adv. and Appl., 110, 227-234 (1999) | MR | Zbl

[18] Laptev A., Weidl T.: Sharp Lieb-Thirring inequalities in high dimensions. Acta Mathematica, 184, 87-111 (2000) | MR | Zbl

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

[20] Lieb, E.H.: The number of bound states of one body Schrödinger operators and the Weyl problem. Bull. Amer. Math. Soc., 82, 751-753 (1976) | Zbl

[21] Lieb, E.H.: Lieb-Thirring Inequalities. Preprint mp-arc 00-132 (2000)

[22] Lieb E.H. and Thirring, W.: 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, 269-303 (1976) | Zbl

[23] Netrusov Y. and Weidl T.: On Lieb-Thirring inequalities for higher order operators with critical and subcritical powers. Comm. Math. Phys., 182 (1), 355-370 (1996) | MR | Zbl

[24] Rozenblum, G.V.: Distribution of the discrete spectrum of singular differential operators. Dokl. AN SSSR, 202, 1012-1015 (1972), Izv. VUZov, Matematika, 1, 75-86 (1976) | MR | Zbl

[25] Ruelle D.: Large volume limit of the distribution of characteristic exponents in turbulence. Comm. Math. Phys., 87, 287-302 (1982) | MR | Zbl

[26] Y.Schwinger: On the bound states for a given potential. Proc. Nat. Acad. Sci. U.S.A., 47, 122-129 (1961) | MR

[27] B. Simon: The bound state of weakly coupled Schrödinger operators on one and two dimensions. Ann. Physics, 97 (2), 279-288, (1976) | MR | Zbl

[28] Weidl, T.: On the Lieb-Thirring constants Lγ,1 for γ ≥ 1/2. Comm. Math. Phys., 178, 135-146 (1996) | MR | Zbl

[29] Weidl, T.: Remarks on virtual bound states for semi-bounded operators. Comm. Part. Diff. Equ., 24 (1&2), 25-60, (1999) | MR | Zbl