Schrödinger Operator with Non-Zero Accumulation Points of Complex Eigenvalues

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.

This is a preview of subscription content, log in to check access.

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)

    ADS  MathSciNet  Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google Scholar 

  7. 7

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

    MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

  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)

  13. 13

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

    ADS  Article  MATH  Google 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)

  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)

  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)

  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)

  18. 18

    Wang X.P.: Number of eigenvalues for dissipative Schrödinger operators under perturbation. J. Math. Pure Appl. (9) 96(5), 409–422 (2011)

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sabine Bögli.

Additional information

Communicated by R. Seiringer

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bögli, S. Schrödinger Operator with Non-Zero Accumulation Points of Complex Eigenvalues. Commun. Math. Phys. 352, 629–639 (2017). https://doi.org/10.1007/s00220-016-2806-5

Download citation