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Annales Henri Poincaré

, Volume 20, Issue 7, pp 2447–2479 | Cite as

\(L_p\)-Spectrum and Lieb–Thirring Inequalities for Schrödinger Operators on the Hyperbolic Plane

  • Marcel HansmannEmail author
Article
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Abstract

This paper deals with the \(L_p\)-spectrum of Schrödinger operators on the hyperbolic plane. We establish Lieb–Thirring-type inequalities for discrete eigenvalues and study their dependence on p. Some bounds on individual eigenvalues are derived as well.

Notes

References

  1. 1.
    Abramov, A.A., Aslanyan, A., Davies, E.B.: Bounds on complex eigenvalues and resonances. J. Phys. A 34(1), 57–72 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. In: National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)Google Scholar
  3. 3.
    Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction, p. 223. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bögli, S.: Schrödinger operator with non-zero accumulation points of complex eigenvalues. Commun. Math. Phys. 352(2), 629–639 (2017)CrossRefzbMATHGoogle Scholar
  5. 5.
    Borichev, A., Golinskii, L., Kupin, S.: A Blaschke-type condition and its application to complex Jacobi matrices. Bull. Lond. Math. Soc. 41(1), 117–123 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borthwick, D.: Spectral theory of infinite-area hyperbolic surfaces. In: Progress in Mathematics, vol. 256, Birkhäuser Boston Inc, Boston (2007)Google Scholar
  7. 7.
    Calderón, A.-P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davies, E.B.: Heat kernels and spectral theory. In: Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge (1989)Google Scholar
  9. 9.
    Davies, E.B., Simon, B., Taylor, M.: \(L^p\) spectral theory of Kleinian groups. J. Funct. Anal. 78(1), 116–136 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Demuth, M., Hansmann, M., Katriel, G.: On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal. 257(9), 2742–2759 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demuth, M., Hansmann, M., Katriel, G.: Eigenvalues of non-selfadjoint operators: a comparison of two approaches. In: Mathematical Physics, Spectral Theory and Stochastic Analysis, Operator Theory: Advances and Applications, vol. 232, Birkhäuser/Springer Basel AG, Basel, pp. 107–163 (2013)Google Scholar
  12. 12.
    Demuth, M., Hansmann, M., Katriel, G.: Lieb–Thirring type inequalities for Schrödinger operators with a complex-valued potential. Integral Equ. Oper. Theory 75(1), 1–5 (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Demuth, M., Katriel, G.: Eigenvalue inequalities in terms of Schatten norm bounds on differences of semigroups, and application to Schrödinger operators. Ann. Henri Poincaré 9(4), 817–834 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Elstrodt, J.: Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I. Math. Ann. 203, 295–300 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Elstrodt, J.: Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. II. Math. Z. 132, 99–134 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Enblom, A.: Estimates for eigenvalues of Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 106(2), 197–220 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Am. Math. Soc. 370(1), 219–240 (2018)CrossRefzbMATHGoogle Scholar
  19. 19.
    Frank, R.L., Laptev, A., Lieb, E.H., Seiringer, R.: Lieb–Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77(3), 309–316 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Frank, R.L., Laptev, A., Seiringer, R.: A sharp bound on eigenvalues of Schrödinger operators on the halfline with complex-valued potentials. In: Spectral Theory and Analysis, Operator Theory: Advances and Applications, vol. 214, Birkhäuser Verlag, Basel, pp. 39–44 (2011)Google Scholar
  21. 21.
    Frank, R.L., Sabin, J.: Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Am. J. Math. 139(6), 1649–1691 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Frank, R.L., Simon, B.: Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory 7(3), 633–658 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators. Operator Theory: Advances and Applications, vol. 49, Birkhäuser Verlag, Basel (1990)Google Scholar
  24. 24.
    Gohberg, I., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. American Mathematical Society, Providence (1969)zbMATHGoogle Scholar
  25. 25.
    Golinskii, L., Kupin, S.: On complex perturbations of infinite band Schrödinger operators. Methods Funct. Anal. Topol. 21(3), 237–245 (2015)zbMATHGoogle Scholar
  26. 26.
    Hansmann, M.: An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators. Lett. Math. Phys. 98(1), 79–95 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hansmann, M.: Perturbation determinants in Banach spaces—with an application to eigenvalue estimates for perturbed operators. Math. Nachr. 289(13), 1606–1625 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hansmann, M.: Some remarks on upper bounds for Weierstrass primary factors and their application in spectral theory. Complex Anal. Oper. Theory 11(6), 1467–1476 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hebey, E., Robert, F.: Sobolev spaces on manifolds. In: Handbook of Global Analysis, vol. 1213, Elsevier Sci. B. V., Amsterdam, pp. 375–415 (2008)Google Scholar
  30. 30.
    Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in \(L_p({ R}^\nu )\) is \(p\)-independent. Commun. Math. Phys. 104(2), 243–250 (1986)CrossRefzbMATHGoogle Scholar
  31. 31.
    Kato, T.: Perturbation theory for linear operators. In: Classics in Mathematics (Reprint of the 1980 edition), Springer, Berlin (1995)Google Scholar
  32. 32.
    König, H.: Eigenvalue distribution of compact operators. In: Operator Theory: Advances and Applications, vol. 16, Birkhäuser Verlag, Basel (1986)Google Scholar
  33. 33.
    König, H., Retherford, J., Tomczak-Jaegermann, N.: On the eigenvalues of \((p,\,2)\)-summing operators and constants associated with normed spaces. J. Funct. Anal. 37(1), 88–126 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kunstmann, P.C.: Heat kernel estimates and \(L^p\) spectral independence of elliptic operators. Bull. Lond. Math. Soc. 31(3), 345–353 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Laptev, A., Safronov, O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys. 292(1), 29–54 (2009)CrossRefzbMATHGoogle Scholar
  36. 36.
    Levin, D., Solomyak, M.: The Rozenblum–Lieb–Cwikel inequality for Markov generators. J. Anal. Math. 71, 173–193 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Lieb, E.H., Thirring, W.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975)CrossRefGoogle Scholar
  38. 38.
    Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to sobolev inequalities. In: Essays in Honor of Valentine Bargmann, Studies in Mathematical Physics, Princeton (1976)Google Scholar
  39. 39.
    Liskevich, V., Vogt, H.: On \(L^p\)-spectra and essential spectra of second-order elliptic operators. Proc. Lond. Math. Soc. 80(3), 590–610 (2000)CrossRefzbMATHGoogle Scholar
  40. 40.
    Ouhabaz, E.M., Poupaud, C.: Remarks on the Cwikel–Lieb–Rozenblum and Lieb–Thirring estimates for Schrödinger operators on Riemannian manifolds. Acta Appl. Math. 110(3), 1449–1459 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Pietsch, A.: Eigenvalues and \(s\)-numbers. Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 43, Akademische Verlagsgesellschaft Geest & Portig K.-G, Leipzig (1987)Google Scholar
  42. 42.
    Pietsch, A., Triebel, H.: Interpolationstheorie für Banachideale von beschränkten linearen Operatoren. Stud. Math. 31, 95–109 (1968)CrossRefzbMATHGoogle Scholar
  43. 43.
    Pommerenke, Ch.: Boundary behaviour of conformal maps. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer, Berlin (1992)Google Scholar
  44. 44.
    Safronov, O.: Estimates for eigenvalues of the Schrödinger operator with a complex potential. Bull. Lond. Math. Soc. 42(3), 452–456 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Safronov, O.: On a sum rule for Schrödinger operators with complex potentials. Proc. Am. Math. Soc. 138(6), 2107–2112 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7(3), 447–526 (1982)CrossRefzbMATHGoogle Scholar
  47. 47.
    Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton (1971)Google Scholar
  49. 49.
    Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Sturm, K.-T.: On the \(L^p\)-spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118(2), 442–453 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Taylor, M.E.: \(L^p\)-estimates on functions of the Laplace operator. Duke Math. J. 58(3), 773–793 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Terras, A.: Harmonic Analysis on Symmetric Spaces–Euclidean Space, the Sphere, and the Poincaré Upper Half-plane, 2nd edn. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  53. 53.
    Triebel, H.: Theory of function spaces. II. In: Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel (1992)Google Scholar

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Authors and Affiliations

  1. 1.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany

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