On General Cwikel–Lieb–Rozenblum and Lieb–Thirring Inequalities

Chapter
Part of the International Mathematical Series book series (IMAT, volume 13)

Abstract

These classical inequalities allow one to estimate the number of negative eigenvalues and the sums \(S_\gamma = \sum { |\lambda _i |^\gamma } \) for a wide class of Schrödinger operators. We provide a detailed proof of these inequalities for operators on functions in metric spaces using the classical Lieb approach based on the Kac–Feynman formula. The main goal of the paper is a new set of examples which include perturbations of the Anderson operator, operators on free, nilpotent, and solvable groups, operators on quantum graphs, Markov processes with independent increments. The study of the examples requires an exact estimate of the kernel of the corresponding parabolic semigroup on the diagonal. In some cases the kernel decays exponentially as t → ∞. This allows us to consider very slow decaying potentials and obtain some results that are precise in the logarithmical scale.

Keywords

Manifold Boris Rene 

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References

  1. 1.
    Birman, M., Solomyak, M.: Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations. Adv. Sov. Math. 7 (1991)Google Scholar
  2. 2.
    Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operator. Birhauser, Basel etc. (1990)Google Scholar
  3. 3.
    Chen, K., Molchanov, S., Vainberg, B.: Localization on Avron–Exner–Last graphs: I. Local perturbations. Contemp. Math. 415, 81–92 (2006)MathSciNetGoogle Scholar
  4. 4.
    Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Daubechies, I.: An uncertanty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90, 511–520 (1983)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of the certain Markov process expectations for large time I.II. Commun. Pure Appl. Math. 28, 1–47 (1975)MATHMathSciNetGoogle Scholar
  7. 7.
    Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Commun. Pure Appl. Math. 28, no. 4, 525–565 (1975)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Eisenhart, L. P.: Rimannian Geometry. Princeton Univ. Press, Princeton (1997)Google Scholar
  9. 9.
    Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents. Acta Math. 139, no. 1, 95–153 (1977)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gikhman, I., Skorokhod, A.: Introduction to the Theory of Random Processes, Dover Publ. Inc., Mineola, NY (1996)Google Scholar
  11. 11.
    Guillotin-Plantard, N., Rene Schott: Dynamic random walks on Heisenberg groups. J. Theor. Probab. 19, no. 2, 377–395 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Karpelevich, F.I., Tutubalin, V.N., Shur, M.G.: Limit theorems for the compositions of distributions in the Lobachevsky plane and space. Theor. Probab. Appl. 4, 399–402 (1959)CrossRefGoogle Scholar
  13. 13.
    Konakov, V., Menozzi, S., Molchanov, S.: [In preparation]Google Scholar
  14. 14.
    McKean, H.: Stochastic Integrals. Am. Math. Soc., Providence, RI (2005)MATHGoogle Scholar
  15. 15.
    Lieb, E.: Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Am. Math. Soc. 82, no. 5, 751–753 (1976)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lieb, E.: The number of bound states of one-body Schrödinger operators and the Weyl problem. In: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 241–252 (1980)MathSciNetGoogle Scholar
  17. 17.
    Lieb, E., Thirring, W.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975)CrossRefGoogle Scholar
  18. 18.
    Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, pp. 269–303, Princeton Univ. Press, Princeton, (1976)Google Scholar
  19. 19.
    Maz'ya, V.: Analytic criteria in the qualitative spectral analysis of the Schrödinger operator. In: Spectral Theory and Mathematical Physics: a Festschrift in honor of Barry Simon's 60th birthday, 257–288. Am. Math. Soc., Providence, RI (2007)Google Scholar
  20. 20.
    Maz'ya, V., Shubin, M.: Discreteness of spectrum and positivity criteria for Schrödinger operators. Ann. Math. 162, 1–24 (2005)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Rashevsky, P. K.: Riemannian Geometry and Tensor Analysis (Russian). “Nauka”, Moscow (1967)Google Scholar
  22. 22.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. 4. Acad. Press, N.Y. (1978)MATHGoogle Scholar
  23. 23.
    Rozenblum, G.: Distribution of the discrete spectrum of singular differential operators (Russian). Dokl. Akad. Nauk SSSR 202 1012–1015 (1972); English transl.: Sov. Math. Dokl. 13, 245–249 (1972)Google Scholar
  24. 24.
    Rozenblum, G., Solomyak, M.: LR-estimate for the generators of positivity preserving and positively dominated semigroups (Russian). Algebra Anal. 9, no. 6, 214–236 (1997); English transl.: St. Petersburg Math. J. 9, no. 6, 1195–1211 (1998)Google Scholar
  25. 25.
    Rozenblum, G., Solomyak, M.: Counting Schrödinger boundstates: semiclassics and beyond. In: Maz'ya, V. (ed.), Sobolev Spaces in Mathematics. II: Applications in Analysis and Parrtial Differential Equations. Springer, New York; Tamara Rozhkovskaya Publisher, Novosibirsk. International Mathematical Series 9, 329–354 (2009)Google Scholar
  26. 26.
    Yor, M.: On some exponential functionals of brownian motion. Adv. Appl. Prob. 24, 509–531 (1992)MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA

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