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Trace Formulas for Schrödinger Operators with Complex Potentials

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Abstract

We consider 3-dimensonal Schrödinger operators with complex potential. We obtain new trace formulas with new terms, associated with singular measure. In order to prove these results, we study analytic properties of a modified Fredholm determinant as a function from Hardy spaces in the upper half-plane. In fact, we reformulate spectral theory problems as problems of analytic functions from Hardy spaces.

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References

  1. O. Blumenthal, “Einige Anwendungen der Sehnen- und Tangententrapezformeln,” Christiaan Huygens 3, 1–17 (1923).

    MATH  Google Scholar 

  2. V. S. Buslaev, “The Trace Formulae and Certain Asymptotic Estimates of the Kernel of the Resolvent for the Schrödinger Operator in Three-Dimensional Space,” Probl. Math. Phys. No. I, Spectral Theory and Wave Processes, 82–101 (1966), Izdat. Leningrad Univ., Leningrad.

    Google Scholar 

  3. Y. Colin de Verdière, “Une formule de traces pour l’opérateur de Schrödinger dans ℝ3,” Ann. Sci. École Norm. Sup. 14, 27–39 (1981).

    Article  MathSciNet  Google Scholar 

  4. E. B. Davies, “J. Nath, Schrödinger operators with Slowly Decaying Potentials,” J. Comput. Appl. Math. 148, 1–28 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  5. M. Demuth, M. Hansmann, and G. Katriel, “On the Discrete Spectrum of Nonself-Adjoint Operators,” J. Funct. Anal. 257(9), 2742–2759 (2009).

    Article  MathSciNet  Google Scholar 

  6. M. Demuth, M. Hansmann, and G. Katriel, “Eigenvalues of Nonself-Adjoint Operators: A Comparison of Two Approaches,” Math. Phys. Spectral Theory and Stochastic Analysis, Springer, 2013, 107–163.

  7. A. Denjoy, “Sur les produits canoniques dordre infini,” J. de Math. 6(6), 1–136 (1910).

    MATH  Google Scholar 

  8. R. L. Frank, “Eigenvalue Bounds for Schrödinger Operators with Complex Potentials,” Bull. Lond. Math. Soc. 43, 745–750 (2011).

    Article  MathSciNet  Google Scholar 

  9. R. L. Frank, “Eigenvalue Bounds for Schrödinger Operators with Complex Potentials. III,” Trans. Amer. Math. Soc. 370(1), 219–240 (2018).

    Article  MathSciNet  Google Scholar 

  10. R. L. Frank, A. Laptev, E. H. Lieb, and R. Seiringer, “Lieb-Thirring Inequalities for Schrödinger Operators with Complex-Valued Potentials,” Lett. Math. Phys. 77, 309–316 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  11. R. L. Frank, A. Laptev, and O. Safronov, “On the Number of Eigenvalues of Schrödinger Operators with Complex Potentials,” J. Lond. Math. Soc. 94(2), 377–390 (2016).

    Article  MathSciNet  Google Scholar 

  12. R. L. Frank and J. Sabin, “Restriction Theorems for Orthonormal Functions, Strichartz Inequalities, and Uniform Sobolev Estimates,” Amer. J. Math. 139(6), 1649–1691 (2017).

    Article  MathSciNet  Google Scholar 

  13. J. Garnett, Bounded Analytic Functions (Academic Press, New York, London, 1981).

    MATH  Google Scholar 

  14. L. Golinskii and S. Kupin, “On Complex Perturbations of Infinite Band Schrödinger Operators,” Methods Funct. Anal. Topology 21(3), 237–245 (2015).

    MathSciNet  MATH  Google Scholar 

  15. L. Guillopé, “Asymptotique de la phase de diffusion pour l’opérateur de Schrödinger dans ℝn,” Séminaire E.D.P., 1984–1985, Exp. No. V, Ecole Polytechnique, 1985.

  16. I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonself-Adjoint Operators (Translated from the Russian, Transl. Math. Monogr., Vol. 18 AMS, Providence, R.I., 1969).

    Google Scholar 

  17. M. Hansmann, “Some Remarks on Upper Bounds for Weierstrass Primary Factors and Their Application in Spectral Theory,” Complex Anal. Oper. Theory 11(6), 1467–1476 (2017).

    Article  MathSciNet  Google Scholar 

  18. H. Isozaki and E. Korotyaev, “Inverse Problems, Trace Formulae for Discrete Schröodinger Operators,” Ann. Henri Poincaré 13(4), 751–788 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  19. A. Jensen, High Energy Asymptotics for the Total Scattering Phase in Potential Scattering Theory Functional-analytic methods for partial differential equations (Tokyo, 1989), 187195

  20. A. Jensen, Lect. Notes in Math., 1450 (Springer, Berlin, 1990).

    Google Scholar 

  21. A. Jensen, “High Energy Resolvent Estimates for Schröodinger Operators,” Ideas and methods in quantum and statistical physics (Oslo, 1988), 254–260, (Cambridge Univ. Press, Cambridge, 1992).

    Google Scholar 

  22. P. Koosis, Introduction to HpSpaces, Vol. 115 (Cambridge Tracts in Mathematic, 1998).

  23. E. Korotyaev, “Trace Formulas for Schrödinger Operators on Lattice,” preprint, arXiv:1702.01388.

  24. E. Korotyaev, “1D Schrödinger Operators with Complex Potentials,” preprint, arXiv:1909.08454.

  25. E. Korotyaev, “Trace Formulas for Schröodinger Operators with Complex Potentials on Half-Line,” to be published in Lett. Math. Phys..

  26. E. Korotyaev and A. Laptev, “Trace Formulae for Schrödinger Operators with Complex-Valued Potentials on Cubic Lattices,” Bull. Math. Sci. 8, 453–475 (2018).

    Article  MathSciNet  Google Scholar 

  27. E. Korotyaev and A. Pushnitski, “A Trace Formula and High-Energy Spectral Asymptotics for the Perturbed Landau Hamiltonian,” J. Funct. Anal. 217(1), 221–248 (2004).

    Article  MathSciNet  Google Scholar 

  28. E. Korotyaev and A. Pushnitski, “Trace Formulae and High Energy Asymptotics for the Stark Operator,” Comm. Partial Differential Equations 28(3–4), 817–842 (2003).

    Article  MathSciNet  Google Scholar 

  29. E. Korotyaev and O. Safronov, “Stark Operators with Complex Potentials,” to be published in Trans. Amer. Math. Soc.

  30. A. Laptev and O. Safronov, “Eigenvalue Estimates for Schrödinger Operators with Complex Potentials,” Comm. Math. Phys. 292(1), 29–54 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  31. E. Lieb and M. Loss, Analysis (AMS, Graduate Studies in Math., Vol. 14, 1997).

  32. E. Lieb and W. Thirring, Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities (Stud. Math. Phys., Princeton University Press, Princeton, 1976), pp. 269–303.

    MATH  Google Scholar 

  33. M. Malamud and H. Neidhardt, “Trace Formulas for Additive and Non-Additive Perturbations,” Adv. Math. 274, 736–832 (2015).

    Article  MathSciNet  Google Scholar 

  34. G. Popov, “Asymptotic Behaviour of the Scattering Phase for the Schröodinger Operator,” C. R. Acad. Bulgare Sci. 35(2), 885–888 (1982).

    MathSciNet  MATH  Google Scholar 

  35. D. Robert, “Asymptotique à grande ènergie de la phase de diffusion pour un potentiel,” Asymptot. Anal. 3, 301–320 (1991).

    Article  Google Scholar 

  36. O. Safronov, “Estimates for Eigenvalues of the Schroödinger Operator with a Complex Potential,” Bull. Lond. Math. Soc. 42(3), 452–456 (2010).

    Article  MathSciNet  Google Scholar 

  37. O. Safronov, “On a Sum Rule for Schröodinger Operators with Complex Potentials,” Proc. Amer. Math. Soc. 138(6), 2107–2112 (2010).

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The author is grateful to Ari Laptev for stimulating discussions about the Schrödinger operators with complex potentials. He is also grateful to Alexei Alexandrov (St. Petersburg) for stimulating discussions and useful comments about Hardy spaces.

Funding

Our study was supported by the RSF grant No. 18-11-00032.

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Authors and Affiliations

  1. Department of Math. Analysis, Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia

    E. Korotyaev

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  1. E. Korotyaev
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Correspondence to E. Korotyaev.

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Korotyaev, E. Trace Formulas for Schrödinger Operators with Complex Potentials. Russ. J. Math. Phys. 27, 82–98 (2020). https://doi.org/10.1134/S1061920820010082

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