Advertisement

Communications in Mathematical Physics

, Volume 352, Issue 2, pp 629–639 | Cite as

Schrödinger Operator with Non-Zero Accumulation Points of Complex Eigenvalues

  • Sabine Bögli
Article

Abstract

We study Schrödinger operators \({H=-\Delta + V}\) in \({L^{2}(\Omega)}\) where \({\Omega}\) is \({\mathbb{R}^d}\) or the half-space \({{\mathbb {R}_{+}^{d}}}\), subject to (real) Robin boundary conditions in the latter case. For \({p > d}\) we construct a non-real potential \({V \in L^{p}(\Omega) \cap L^{\infty}(\Omega)}\) that decays at infinity so that H has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum \({\sigma_{\rm ess}(H)=[0,\infty)}\). This demonstrates that the Lieb–Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol. 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, US Government Printing Office, Washington, DC (1964)Google Scholar
  3. 3.
    Bögli, S.: Convergence of sequences of linear operators and their spectra. arXiv:1604.07732 (2016)
  4. 4.
    Bögli, S.: Local convergence of spectra and pseudospectra. to appear in J. Spectr. Theory. arXiv:1605.01041 (2016)
  5. 5.
    Demuth M., Hansmann M., Katriel G.: On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal. 257(9), 2742–2759 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Demuth M., Hansmann M., Katriel G.: Lieb–Thirring type inequalities for Schrödinger operators with a complex-valued potential. Integral Equ. Oper. Theory 75(1), 1–5 (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Frank R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)MathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frank R.L., Laptev A., Safronov O.: On the number of eigenvalues of Schrödinger operators with complex potentials. J. Lond. Math. Soc. (2) 94(2), 377–390 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frank, R.L., Simon, B.: Eigenvalue bounds for Schrödinger operators with complex potentials. II. to appear in J. Spectr. Theory. arXiv:1504.01144
  11. 11.
    Kato, T.: Perturbation theory for linear operators. Springer, Berlin (Reprint of the 1980 edition) (1995)Google Scholar
  12. 12.
    Laptev, A.: Spectral inequalities for partial differential equations and their applications. In Fifth International Congress of Chinese Mathematicians. Part 1, 2, vol. 2 of AMS/IP Stud. Adv. Math., 51, pt. 1. Am. Math. Soc., Providence, RI, pp. 629–643(2012)Google Scholar
  13. 13.
    Laptev A., Safronov O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys 292(1), 29–54 (2009)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Stud. Math. Phys, 269–303 (1976)Google Scholar
  15. 15.
    Pavlov, B.S.: On a non-selfadjoint Schrödinger operator. In: Probl. Math. Phys., No. 1, Spectral Theory and Wave Processes (Russian). Izdat. Leningrad. Univ., Leningrad, pp. 102–132 (1966)Google Scholar
  16. 16.
    Pavlov, B.S.: On a non-selfadjoint Schrödinger operator. II. In: Problems of Math. Phys., No. 2, Spectral Theory, Diffraction Problems (Russian). Izdat. Leningrad. Univ., Leningrad, pp. 133–157 (1967)Google Scholar
  17. 17.
    Pavlov, B.S.: On a nonselfadjoint Schrödinger operator. III. In: Problems of Math. Phys., No. 3, Spectral Theory (Russian). Izdat. Leningrad. Univ., Leningrad, pp. 59–80 (1968)Google Scholar
  18. 18.
    Wang X.P.: Number of eigenvalues for dissipative Schrödinger operators under perturbation. J. Math. Pure Appl. (9) 96(5), 409–422 (2011)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany

Personalised recommendations