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Optimal Hardy inequalities for Schrödinger operators on graphs

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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.

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References

  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)

  2. Balinsky, A.A., Evans, W.D., Lewis, R.T.: The analysis and geometry of Hardy’s inequality. Universitext. Springer, Cham (2015)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogdan K., Dyda B.: The best constant in a fractional Hardy inequality. Math. Nachr. 284(5-6), 629–638 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonnefont M., Golénia S.: Essential spectrum and Weyl asymptotics for discrete Laplacians. Ann. Fac. Sci. Toulouse Math. (6) 24(3), 563–624 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Devyver B., Pinchover Y.: Optimal L p Hardy-type inequalities. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(1), 93–118 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  10. Frank R.L., Seiringer R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  ADS  MATH  Google Scholar 

  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)

  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)

    Article  ADS  MATH  Google Scholar 

  15. Keller, M., Lenz, D., Schmidt, M., Schwarz, M.: Boundary representation of Dirichlet forms on discrete spaces. arXiv:1711.08304.

  16. Keller, M., Pinchover, Y., Pogorzelski, F.: An improved discrete Hardy inequality. Am. Math. Mon. arXiv:1612.05913

  17. Keller, M., Pinchover, Y., Pogorzelski, F.: Criticality theory for Schrödinger operators on graphs. arXiv:1708.09664

  18. Kovařík H., Laptev A.: Hardy inequalities for Robin Laplacians. J. Funct. Anal. 262(12), 4972–4985 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kufner A., Maligranda L., Persson L.-E.: The prehistory of the Hardy inequality. Am. Math. Mon. 113(8), 715–732 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Loss M., Sloane C.: Hardy inequalities for fractional integrals on general domains. J. Funct. Anal. 259(6), 1369–1379 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  22. Opic, B., Kufner, A.: Hardy-type inequalities, volume 219 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow (1990)

  23. Uchiyama K.: Green’s functions for random walks on Z N. Proc. Lond. Math. Soc. (3) 77(1), 215–240 (1998)

    Article  Google Scholar 

  24. Yafaev D.: Sharp constants in the Hardy–Rellich inequalities. J. Funct Anal. 168(1), 121–144 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Woess, W.: Random walks on infinite graphs and groups, volume 138 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)

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Authors and Affiliations

  1. Institut für Mathematik, Universität Potsdam, 14476, Potsdam, Germany

    Matthias Keller

  2. Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Yehuda Pinchover

  3. Institut für Mathematik, Universität Leipzig, 04109, Leipzig, Germany

    Felix Pogorzelski

Authors
  1. Matthias Keller
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  2. Yehuda Pinchover
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  3. Felix Pogorzelski
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Corresponding author

Correspondence to Felix Pogorzelski.

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Communicated by R. Seiringer

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Keller, M., Pinchover, Y. & Pogorzelski, F. Optimal Hardy inequalities for Schrödinger operators on graphs. Commun. Math. Phys. 358, 767–790 (2018). https://doi.org/10.1007/s00220-018-3107-y

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