Skip to main content
SpringerLink
Log in

Schrödinger Operators with Few Bound States

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, H, with bounded positive ground states: Given a potential V, if both H±V are bounded from below by the ground-state energy of H, then V≡0.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agmon, S.: On positive solutions of elliptic equations with periodic coefficients in ℝn, spectral results and extensions to elliptic operators on Riemannian manifolds. In: Differential Equations (Birmingham, Ala., 1983), North-Holland Math. Stud. 92, Amsterdam: North-Holland, 1984, pp. 7–17

  2. Behncke, H.: Absolute continuity of Hamiltonians with von Neumann Wigner potentials. II. Manuscripta Math. 71, 163–181 (1991)

    Google Scholar 

  3. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Vol. I. New York: Interscience Publishers, Inc., 1953

  4. Damanik, D., Hundertmark, D., Killip, R., Simon, B.: Variational estimates for discrete Schrödinger operators with potentials of indefinite sign. Commun. Math. Phys. 238, 545–562 (2003)

    Article  Google Scholar 

  5. Damanik, D., Hundertmark, D., Simon, B.: Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators. J. Funct. Anal. 205, 357–379 (2003)

    Article  Google Scholar 

  6. Damanik, D., Killip, R.: Half-line Schrödinger operators with no bound states. Acta Math. 193, 31–72 (2004)

    Google Scholar 

  7. Damanik, D., Killip, R., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals. III. α-continuity. Commun. Math. Phys. 212, 191–204 (2000)

    Google Scholar 

  8. Damanik, D., Remling, C.: Schrödinger operators with many bound states, Preprint (2004)

  9. Gilbert, D.J.: On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints. Proc. Roy. Soc. Edinburgh Sect. A 112, 213–229 (1989)

    Google Scholar 

  10. Gilbert, D.J., Pearson, D.B.: On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128, 30–56 (1987)

    Article  Google Scholar 

  11. Hundertmark, D., Simon, B.: Lieb-Thirring inequalities for Jacobi matrices. J. Approx. Theory 118, 106–130 (2002)

    Article  Google Scholar 

  12. Jacobi, C.: Zur Theorie der Variations-Rechnung und der Differential-Gleichungen. J. Reine Angew. Math. 17, 68–82 (1837)

    Google Scholar 

  13. Jitomirskaya, S., Last, Y.: Power-law subordinacy and singular spectra. I. Half-line operators. Acta Math. 183, 171–189 (1999)

    Google Scholar 

  14. Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. of Math. 158, 253–321 (2003)

    Google Scholar 

  15. Kirsch, W., Simon, B.: Corrections to the classical behavior of the number of bound states of Schrödinger operators. Ann. Physics 183, 122–130 (1988)

    Article  Google Scholar 

  16. Naiman, P.B.: The set of isolated points of increase of the spectral function pertaining to a limit-constant Jacobi matrix (Russian). Izv. Vysš. Učebn. Zaved. Mat. 1(8), 129–135 (1959)

    Google Scholar 

  17. Peherstorfer, F., Yuditskii, P.: Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. Proc. Amer. Math. Soc. 129, 3213–3220 (2001)

    Article  Google Scholar 

  18. Pinchover, Y.: On positive solutions of second-order elliptic equations, stability results, and classification. Duke Math. J. 57, 955–980 (1988)

    Article  Google Scholar 

  19. Pinchover, Y.: On criticality and ground states of second order elliptic equations. II. J. Differ. Eqs. 87, 353–364 (1990)

    Article  Google Scholar 

  20. Pinsky, R.G.: Second order elliptic operators with periodic coefficients: criticality theory, perturbations, and positive harmonic functions. J. Funct. Anal. 129, 80–107 (1995)

    Article  Google Scholar 

  21. Safronov, O.: Multi-dimensional Schrödinger operators with no negative spectrum. http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=04-273, 2004

  22. Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators. Proc. Amer. Math. Soc. 124, 3361–3369 (1996)

    Article  Google Scholar 

  23. Simon, B.: The Golinskii-Ibragimov method and a theorem of Damanik and Killip. Int. Math. Res. Not. 2003, 1973–1986

  24. Simon, B.: Schrödinger semigroups. Bull. Amer. Math. Soc. 7, 447–526 (1982)

    Google Scholar 

  25. Simon, B., Zlatoš, A.: Sum rules and the Szegő condition for orthogonal polynomials on the real line. Commun. Math. Phys. 242, 393–423 (2003)

    Google Scholar 

  26. Stolz, G.: Bounded solutions and absolute continuity of Sturm-Liouville operators. J. Math. Anal. Appl. 169, 210–228 (1992)

    Article  Google Scholar 

  27. Tamura, H.: The Efimov effect of three-body Schrödinger operators. J. Funct. Anal. 95, 433–459 (1991)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics 253–37, California Institute of Technology, Pasadena, CA, 91125, USA

    David Damanik & Barry Simon

  2. Department of Mathematics, University of California, Los Angeles, CA, 90055, USA

    Rowan Killip

Authors
  1. David Damanik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rowan Killip
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Barry Simon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Damanik.

Additional information

Communicated by M. Aizenman

D. D. was supported in part by NSF grant DMS–0227289.

R. K. was supported in part by NSF grant DMS–0401277.

B. S. was supported in part by NSF grant DMS–0140592.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Damanik, D., Killip, R. & Simon, B. Schrödinger Operators with Few Bound States. Commun. Math. Phys. 258, 741–750 (2005). https://doi.org/10.1007/s00220-005-1366-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1366-x

Keywords

Search

Navigation