Communications in Mathematical Physics

, Volume 326, Issue 2, pp 531–541 | Cite as

On Some Sharp Spectral Inequalities for Schrödinger Operators on Semiaxis

  • Pavel Exner
  • Ari Laptev
  • Muhammad Usman


In this paper we obtain sharp Lieb–Thirring inequalities for a Schrödinger operator on semiaxis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and spectral inequalities for Schrödinger operators on half-spaces with Robin boundary conditions.


Ground State Energy Neumann Boundary Condition Negative Eigenvalue Robin Boundary Condition Star Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AizL.
    Aizenman M., Lieb E.H.: On semi-classical bounds for eigenvalues of Schrödinger operators. Phys. Lett. 66, 427–429 (1978)CrossRefMathSciNetGoogle Scholar
  2. BT.
    Boumenir A., Tuan V.K.: A trace formula and Schmincke inequality on the half-line. Proc. Amer. Math. Soc. 137(3), 1039–1049 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  3. BL.
    Benguria R., Loss M.: A simple proof of a theorem by Laptev and Weidl. Math. Res. Lett. 7(2–3), 195–203 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  4. Con.
    Conlon J.G.: A new proof of the Cwikel–Lieb–Rosenbljum bound. Rocky Mountain J. Math. 15, 117–122 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  5. Cw.
    Cwikel M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Trans. AMS 224, 93–100 (1977)MathSciNetGoogle Scholar
  6. DLL.
    Dolbeault J., Laptev A., Loss M.: Lieb–Thirring inequalities with improved constants. JEMS 10, 1121–1126 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. Fe.
    Fefferman C.L.: The uncertainty principle. Bull. Amer. Math. Soc. 9(2), 129–206 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  8. Fr.
    Frank, R.L.: Cwikel’s theorem and the CLR inequality. Accepted by JST, Available at [math.sp], 2012
  9. FL.
    Frank, R.L., Laptev, A.: Spectral inequalities for Schrödinger operators with surface potentials. In: Spectral theory of differential operators, Amer. Math. Soc.Transl. Ser. 2, 225, Providence, RI: Amer. Math. Soc., 2008, pp. 91–102Google Scholar
  10. HLW.
    Hundertmark D., Laptev A., Weidl T.: New bounds on the Lieb–Thirring constants. Inv. Math. 140, 693–704 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. HLT.
    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, 719–731 (1998)zbMATHMathSciNetGoogle Scholar
  12. Ku.
    Kuchment P.: Quantum graphs: I. Some basic structures. Waves in Random Media 14, S107–S128 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. L1.
    Laptev A.: Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces. J. Funct. Anal. 151, 531–545 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  14. LW1.
    Laptev A., Weidl T.: Sharp Lieb–Thirring inequalities in high dimensions. Acta Mathematica 184, 87–111 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. LW2.
    Laptev, A., Weidl, T.: Recent results on Lieb–Thirring inequalities. Journées Équations aux Dérivées Partielles? (La Chapelle sur Erdre, 2000), Exp. No. XX, Nantes: Univ. Nantes, 2000Google Scholar
  16. LY.
    Li P., Yau S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. L.
    Lieb, E.H.: The Number of Bound States of One-Body Schrödinger Operators and the Weyl Problem. In: Proceedings of the Amer. Math. Soc. Symposia in Pure Math. 36, Providence, RI: Amer. Math. Soc., 1980, pp. 241–252Google Scholar
  18. LT.
    Lieb, E.H., 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, NJ: Princeton Univ. Press, 1976, pp. 269–303Google Scholar
  19. Roz.
    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)Google Scholar
  20. S.
    Schmincke U.W.: On Schrödingers factorization method for Sturm–Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 80, 67–84 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  21. W1.
    Weidl T.: On the Lieb–Thirring constants L γ,1 for γ ≥  1/2. Commun. Math. Phys. 178, 135–146 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Doppler Institute for Mathematical Physics and Applied MathematicsPragueCzechia
  2. 2.Nuclear Physics Institute ASCRŘež near PragueCzechia
  3. 3.Imperial College LondonLondonUK
  4. 4.Institut de Mathématiques de BordeauxUniversité Bordeaux 1TalenceFrance

Personalised recommendations