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Improved Eigenvalue Bounds for Schrödinger Operators with Slowly Decaying Potentials

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Abstract

We extend a result of Davies and Nath (J Comput Appl Math 148(1):1–28, 2002) on the location of eigenvalues of Schrödinger operators with slowly decaying complex-valued potentials to higher dimensions. In this context, we also discuss various examples related to the Laptev and Safronov conjecture (Laptev and Safronov in Commun Math Phys 292(1):29–54, 2009).

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Notes

  1. Note that \(\Vert V_n^{\frac{1}{2}}\psi _n\Vert _2\) is finite in view of Sobolev embedding.

  2. Similarly, the previous example could also be regarded as a simplified one-dimensional version.

References

  1. Abramov, A.A., Aslanyan, A., Davies, E.B.: Bounds on complex eigenvalues and resonances. J. Phys. A 34(1), 57–72 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  2. Bögli, S.: Schrödinger operator with non-zero accumulation points of complex eigenvalues. Commun. Math. Phys. 352(2), 629–639 (2017)

    Article  ADS  Google Scholar 

  3. Cuenin, J.-C.: Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials. J. Funct. Anal. 272(7), 2987–3018 (2017)

    Article  MathSciNet  Google Scholar 

  4. Cuenin, J.-C.: Embedded eigenvalues for generalized Schrödinger operators, 18 pages (2017). arXiv:1709.06989

  5. Davies, E.B., Nath, J.: Schrödinger operators with slowly decaying potentials. J. Comput. Appl. Math., 148(1):1–28 (2002) (on the occasion of the 65th birthday of Professor Michael Eastham)

    Article  ADS  MathSciNet  Google Scholar 

  6. Demuth, M., Hansmann, M., Katriel, G.: On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal. 257(9), 2742–2759 (2009)

    Article  MathSciNet  Google Scholar 

  7. Demuth, M., Hansmann, M., Katriel, G.: Eigenvalues of non-selfadjoint operators: a comparison of two approaches. In: Mathematical Physics, Spectral Theory and Stochastic Analysis, volume 232 of Operator Theory Advances Applications. Birkhäuser/Springer Basel AG, Basel, pp. 107–163 (2013)

    Chapter  Google Scholar 

  8. Dyatlov, S., Zworski, M.: Mathematical theory of scattering resonances. Graduate Studies in Mathematics, vol. 200. American Mathematical Society (2019)

  9. Fanelli, L., Krejčiřík, D., Vega, L.: Absence of eigenvalues of two-dimensional magnetic Schrödinger operators. J. Funct. Anal. 275(9), 2453–2472 (2018)

    Article  MathSciNet  Google Scholar 

  10. Fanelli, L., Krejčiřík, D., Vega, L.: Spectral stability of Schrödinger operators with subordinated complex potentials. J. Spectr. Theory 8(2), 575–604 (2018)

    Article  MathSciNet  Google Scholar 

  11. Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)

    Article  MathSciNet  Google Scholar 

  12. Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Am. Math. Soc. 370(1), 219–240 (2018)

    Article  Google Scholar 

  13. 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)

    Article  ADS  MathSciNet  Google Scholar 

  14. Frank, R.L., Laptev, A., Safronov, O.: On the number of eigenvalues of Schrödinger operators with complex potentials. ArXiv e-prints, Jan. (2016)

    Article  MathSciNet  Google Scholar 

  15. Frank, R.L., Sabin, J.: Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Am. J. Math. 139(6), 1649–1691 (2017)

    Article  MathSciNet  Google Scholar 

  16. Frank, R.L., Simon, B.: Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory 7(3), 633–658 (2017)

    Article  MathSciNet  Google Scholar 

  17. Guillarmou, C., Hassell, A., Krupchyk, K.: Eigenvalue bounds for non-self-adjoint Schrödinger operators with non-trapping metrics. ArXiv e-prints, Sept (2017)

  18. 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(4), 719–731 (1998)

    Article  MathSciNet  Google Scholar 

  19. Ionescu, A.D., Jerison, D.: On the absence of positive eigenvalues of Schrödinger operators with rough potentials. Geom. Funct. Anal. 13(5), 1029–1081 (2003)

    Article  MathSciNet  Google Scholar 

  20. Kenig, C.E., Ruiz, A., Sogge, C.D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55(2), 329–347 (1987)

    Article  MathSciNet  Google Scholar 

  21. Laptev, A., Safronov, O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys. 292(1), 29–54 (2009)

    Article  ADS  Google Scholar 

  22. Lee, Y., Seo, I.: A note on eigenvalue bounds for Schrödinger operators. J. Math. Anal. Appl. 470(1), 340–347 (2019)

    Article  MathSciNet  Google Scholar 

  23. Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the schrodinger hamiltonian and their relation to sobolev inequalities. In: Studies in Mathematical Physics. Princeton University Press (1976)

  24. Mizutani, H.: Eigenvalue bounds for non-self-adjoint Schrödinger operators with the inverse-square potential. ArXiv e-prints, July (2016)

  25. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J. Princeton Mathematical Series, No. 32 (1971)

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Acknowledgements

The author gratefully acknowledges the hospitality of the Institut Mittag-Leffler and the invitation to the thematic program Spectral Methods in Mathematical Physics. The present article was written during the authors stay. The idea to use Lemma 4 and the proof thereof is attributed to Chris Sogge, to whom the author is thankful.

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  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333, Munich, Germany

    Jean-Claude Cuenin

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Correspondence to Jean-Claude Cuenin.

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Cuenin, JC. Improved Eigenvalue Bounds for Schrödinger Operators with Slowly Decaying Potentials. Commun. Math. Phys. 376, 2147–2160 (2020). https://doi.org/10.1007/s00220-019-03635-w

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