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Letters in Mathematical Physics

, Volume 106, Issue 2, pp 197–220 | Cite as

Estimates for Eigenvalues of Schrödinger Operators with Complex-Valued Potentials

  • Alexandra Enblom
Article

Abstract

New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrödinger operators are obtained in terms of L p -norms of the potentials. The results cover and improve those known previously, in particular, due to Frank (Bull Lond Math Soc 43(4):745–750, 2011), Safronov (Proc Am Math Soc 138(6):2107–2112, 2010), Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009). We mention the estimations of the eigenvalues situated in the strip around the real axis (in particular, the essential spectrum). The method applied for this case involves the unitary group generated by the Laplacian. The results are extended to the more general case of polyharmonic operators. Schrödinger operators with slowly decaying potentials and belonging to weak Lebesgue’s classes are also considered.

Keywords

Schrödinger operators polyharmonic operators complex potential estimation of eigenvalues 

Mathematics Subject Classifications

Primary 47F05 Secondary 35P15 81Q12 

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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)CrossRefADSMathSciNetMATHGoogle Scholar
  2. 2.
    Babenko K.I.: An inequality in the theory of Fourier integrals. Izv. Akad. Nauk SSSR Ser. Mat. 25, 531–542 (1961)MathSciNetMATHGoogle Scholar
  3. 3.
    Beckner W.: Inequalities in Fourier analysis. Ann. Math. 2 102(1), 159–182 (1975)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976).(Grundlehren der Mathematischen Wissenschaften, No. 223) Google Scholar
  5. 5.
    Berezin, F.A., Shubin, M.A.: The Schrödinger equation. In: Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991)Google Scholar
  6. 6.
    Davies E.B.: Non-self-adjoint differential operators. Bull. London Math. Soc. 34(5), 513–532 (2002)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Davies E.B., Nath J.: Schrödinger operators with slowly decaying potentials. J. Comput. Appl. Math. 148(1), 1–28 (2002)CrossRefADSMathSciNetMATHGoogle Scholar
  8. 8.
    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)CrossRefADSMathSciNetMATHGoogle Scholar
  9. 9.
    Frank, R.L., Laptev, A., Seiringer, R.: A sharp bound on eigenvalues of Schrödinger operators on the half-line with complex-valued potentials. In: Spectral Theory and Analysis, vol. 214, pp. 39–44. Birkhäuser/Springer Basel AG, Basel (2011); Oper. Theory Adv. Appl.Google Scholar
  10. 10.
    Folland, G.B.: Real analysis. In: Pure and Applied Mathematics (New York), 2nd edn. Wiley, New York (1999)Google Scholar
  11. 11.
    Frank R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007)Google Scholar
  13. 13.
    Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical Society, Providence (1974)Google Scholar
  14. 14.
    Jörgens, K., Weidmann, J.: Spectral properties of Hamiltonian operators. In: Lecture Notes in Mathematics, vol. 313. Springer, Berlin (1973)Google Scholar
  15. 15.
    Kato, T.: Perturbation theory for linear operators. In: Classics in Mathematics. Springer, Berlin (1995). (Reprint of the 1980 edition) Google Scholar
  16. 16.
    Keller J.B.: Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation. J. Math. Phys. 2, 262–266 (1961)CrossRefADSMATHGoogle Scholar
  17. 17.
    Konno R., Konno R.: On the finiteness of perturbed eigenvalues. J. Fac. Sci. Univ. Tokyo Sect. I 13, 55–63 (1966)MathSciNetMATHGoogle Scholar
  18. 18.
    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)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Laptev A., Safronov O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys. 292(1), 29–54 (2009)CrossRefADSMathSciNetMATHGoogle Scholar
  20. 20.
    Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics, pp. 269–303. Princeton University Press, Princeton (1976)Google Scholar
  21. 21.
    O’Neil, R.: Convolution operators and L(pq) spaces. Duke Math. J. 30, 129–142 (1963)Google Scholar
  22. 22.
    Prosser R.T.: Convergent perturbation expansions for certain wave operators. J. Math. Phys. 5, 708–713 (1964)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Rejto P.A.: On partly gentle perturbations. III. J. Math. Anal. Appl. 27, 21–67 (1969)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Safronov O.: Estimates for eigenvalues of the Schrödinger operator with a complex potential. Bull. Lond. Math. Soc. 42(3), 452–456 (2010)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Safronov O.: On a sum rule for Schrödinger operators with complex potentials. Proc. Am. Math. Soc. 138(6), 2107–2112 (2010)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Schechter M.: Essential spectra of elliptic partial differential equations. Bull. Am. Math. Soc. 73, 567–572 (1967)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Schechter, M.: Spectra of partial differential operators. In: North-Holland Series in Applied Mathematics and Mechanics, vol. 14, 2nd edn. North-Holland Publishing Co., Amsterdam (1986)Google Scholar
  28. 28.
    Stummel F.: Singuläre elliptische differential-operatoren in Hilbertschen R äumen. Math. Ann. 132, 150–176 (1956)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden

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