Advertisement

Communications in Mathematical Physics

, Volume 358, Issue 2, pp 767–790 | Cite as

Optimal Hardy inequalities for Schrödinger operators on graphs

  • Matthias Keller
  • Yehuda Pinchover
  • Felix Pogorzelski
Article

Abstract

For a given subcritical discrete Schrödinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H − λw is subcritical in X for all λ < 1, null-critical in X for λ = 1, and supercritical near any neighborhood of infinity in X for any λ > 1. Our results rely on a criticality theory for Schrödinger operators on general weighted graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adimurthi, Yang, Y.: An interpolation of Hardy inequality and Trundinger–Moser inequality in \({\mathbb{R}^N}\) and its applications. Int. Math. Res. Not. IMRN 2010(13), 2394–2426 (2010)Google Scholar
  2. 2.
    Balinsky, A.A., Evans, W.D., Lewis, R.T.: The analysis and geometry of Hardy’s inequality. Universitext. Springer, Cham (2015)Google Scholar
  3. 3.
    Bauer F., Horn P., Lin Y., Lippner G., Mangoubi D., Yau S.-T.: Li–Yau inequality on graphs. J. Differ. Geom. 99(3), 359–405 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bogdan K., Dyda B.: The best constant in a fractional Hardy inequality. Math. Nachr. 284(5-6), 629–638 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bonnefont M., Golénia S.: Essential spectrum and Weyl asymptotics for discrete Laplacians. Ann. Fac. Sci. Toulouse Math. (6) 24(3), 563–624 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Devyver B., Fraas M., Pinchover Y.: Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon. J. Funct. Anal. 266(7), 4422–4489 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Devyver B., Pinchover Y.: Optimal L p Hardy-type inequalities. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(1), 93–118 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dolbeault J., Esteban M.J., Filippas S., Tertikas A.: Rigidity results with applications to best constants and symmetry of Caffarelli–Kohn–Nirenberg and logarithmic Hardy inequalities. Calc. Var. Partial Differ. Equ. 54(3), 2465–2481 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ekholm, T., Frank R.L., Kovařík, H.: Remarks about Hardy inequalities on metric trees. In: Analysis on Graphs and Its Applications, volume 77 of Proceedings of Symposia Pure Mathematics, pp. 369–379. American Mathematical Society, Providence, RI (2008)Google Scholar
  10. 10.
    Frank R.L., Seiringer R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Frank, R.L., Seiringer R.: Sharp fractional Hardy inequalities in half-spaces. In: Laptev, A. (ed) Around the research of Vladimir Maz’ya. I, volume 11 of Int. Math. Ser. (N. Y.), pp. 161–167. Springer, New York (2010)Google Scholar
  12. 12.
    Frank R.L., Simon B., Weidl T.: Eigenvalue bounds for perturbations of Schrödinger operators and Jacobi matrices with regular ground states. Commun. Math. Phys. 282, 199–208 (2008)ADSCrossRefMATHGoogle Scholar
  13. 13.
    Haeseler, S., Keller, M.: Generalized solutions and spectrum for Dirichlet forms on graphs. In: Random walks, boundaries and spectra, volume 64 of Progress in Probability, pp. 181–199. Birkhäuser/Springer Basel AG, Basel (2011)Google Scholar
  14. 14.
    Herbst I.W.: Spectral theory of the operator \({(p^2+m^2)^{1/2}-Ze^2/r }\) . Commun. Math. Phys. 53(3), 285–294 (1977)ADSCrossRefMATHGoogle Scholar
  15. 15.
    Keller, M., Lenz, D., Schmidt, M., Schwarz, M.: Boundary representation of Dirichlet forms on discrete spaces. arXiv:1711.08304.
  16. 16.
    Keller, M., Pinchover, Y., Pogorzelski, F.: An improved discrete Hardy inequality. Am. Math. Mon. arXiv:1612.05913
  17. 17.
    Keller, M., Pinchover, Y., Pogorzelski, F.: Criticality theory for Schrödinger operators on graphs. arXiv:1708.09664
  18. 18.
    Kovařík H., Laptev A.: Hardy inequalities for Robin Laplacians. J. Funct. Anal. 262(12), 4972–4985 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kufner A., Maligranda L., Persson L.-E.: The prehistory of the Hardy inequality. Am. Math. Mon. 113(8), 715–732 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Loss M., Sloane C.: Hardy inequalities for fractional integrals on general domains. J. Funct. Anal. 259(6), 1369–1379 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Naimark K., Solomyak M.: Geometry of Sobolev spaces on regular trees and the Hardy inequalities. Russ. J. Math. Phys. 8(3), 322–335 (2001)MathSciNetMATHGoogle Scholar
  22. 22.
    Opic, B., Kufner, A.: Hardy-type inequalities, volume 219 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow (1990)Google Scholar
  23. 23.
    Uchiyama K.: Green’s functions for random walks on Z N. Proc. Lond. Math. Soc. (3) 77(1), 215–240 (1998)CrossRefGoogle Scholar
  24. 24.
    Yafaev D.: Sharp constants in the Hardy–Rellich inequalities. J. Funct Anal. 168(1), 121–144 (1999)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Woess, W.: Random walks on infinite graphs and groups, volume 138 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael
  3. 3.Institut für MathematikUniversität LeipzigLeipzigGermany

Personalised recommendations