Skip to main content
SpringerLink
Log in

Dynamical Analysis of Schrödinger Operators with Growing Sparse Potentials

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

Abstract

We consider discrete half-line Schrödinger operators H with potentials of the form V(n)=S(n)+Q(n). Here Q is any compactly supported real function, if n=L N and S(n)=0 otherwise, where η ∈ (0,1) and L N is a very fast growing sequence. We study in a rather detailed manner the time-averaged dynamics exp(−itH)ψ for various initial states ψ. In particular, for some ψ we calculate explicitly the “intermittency function” β ψ −(p) which turns out to be nonconstant. The dynamical results obtained imply that the spectral measure of H has exact Hausdorff dimension η for all boundary conditions, improving the result of Jitomirskaya and Last.

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. Combes, J.M., Mantica, G.: Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices. In: Long time behaviour of classical and quantum systems, S. Graffi, A. Martinez (eds.), Series on concrete and appl. math. Vol. 1, Singapore: World Scientific, 2001, pp. 107–123

  2. Damanik, D., Tcheremchantsev, S.: Power-law bounds on transfer matrices and quantum dynamics in one dimension. Commun. Math. Phys. 236, 513–534 (2003)

    MATH  Google Scholar 

  3. del Rio, D., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank-one perturbations, and localization. J. d’Analyse Math. 69, 153–200 (1996)

    Google Scholar 

  4. Gordon, A. Ya.: Deterministic potential with a pure point spectrum. Math. Notes 48, 1197–1203 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Germinet, F., Kiselev, A., Tcheremchantsev, S.: Transfer matrices and transport for Schrödinger operators. Ann. Inst. Fourier 54 (3), 787–830 (2004)

    Google Scholar 

  6. Germinet, F., Klein, A.: Operator kernel estimates for functions of generalized Schrödinger operators. Proc. Amer. Math. Soci. 131, 911–920 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guarneri, I., Schulz-Baldes, H.: Lower bounds on wave-packet propagation by packing dimensions of spectral measure. Math. Phys. Elec. J. 5, 1 (1999)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Killip, R., Kiselev, A., Last, Y.: Dynamical upper bounds on wavepacket spreading. Am. J. Math. 125 (5), 1165–1198 (2003)

    MATH  Google Scholar 

  10. Krutikov, D.: Asymptotics of the Fourier transform of the spectral measure for Schrödinger operators with bounded and unbounded sparse potentials. J. Phys. A 35, 6393–6417 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krutikov, D., Remling, C.: Schrödinger operators with sparse potentials: asymptotics of the Fourier transform of the spectral measure. Commun. Math. Phys. 223, 509–532 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Last, Y.: Quantum dynamics and decompositions of singular continuous spectrum. J. Funct. Anal. 142, 405–445 (1996)

    Article  Google Scholar 

  13. Pearson, D.B.: Singular continuous measures in scattering theory. Commun. Math. Phys. 60, 13–36 (1978)

    MATH  Google Scholar 

  14. Simon, B.: Operators with singular continuous spectrum, VII. Examples with borderline time decay. Commun. Math. Phys. 176, 713–722 (1996)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Simon, B., Spencer, T.: Trace class perturbations and the absence of absolutely continuous spectrum. Commun. Math. Phys. 125, 113–126 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Simon, B., Stolz, G.: Operators with singular continuous spectrum, V. Sparse potentials. Proc. Am. Math. Soc. 124, 2073–2080 (1996)

    Article  MATH  Google Scholar 

  18. Tcheremchantsev, S.: Mixed lower bounds for quantum transport. J. Funct. Anal. 197, 247–282 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zlatos, A.: Sparse potentials with fractional Hausdorff dimension. J. Funct. Anal. 207, 216–252 (2004)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UMR 6628-MAPMO, Université d’Orléans, B.P. 6759, 45067, Orléans Cedex, France

    Serguei Tcheremchantsev

Authors
  1. Serguei Tcheremchantsev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serguei Tcheremchantsev.

Additional information

Communicated by B. Simon

Acknowledgement I would like to thank F. Germinet for useful discussions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tcheremchantsev, S. Dynamical Analysis of Schrödinger Operators with Growing Sparse Potentials. Commun. Math. Phys. 253, 221–252 (2005). https://doi.org/10.1007/s00220-004-1153-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1153-0

Keywords

Search

Navigation