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Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

Abstract

We consider L1L estimates for the time evolution of Hamiltonians H=−Δ+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is ∫(1+|x|)|V(x)|dx<∞, whereas in d=3 it is |V(x)|≤C(1+|x|)−3−.

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References

  1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2), 151–218 (1975)

    Google Scholar 

  2. Artbazar, G., Yajima, K.: The Lp-continuity of wave operators for one dimensional Schrödinger operators. J. Math. Sci. Univ. Tokyo 7(2), 221–240 (2000)

    MATH  Google Scholar 

  3. Deift, P., Trubowitz, E.: Inverse scattering on the line. Comm. Pure Appl. Math. XXXII, 121–251 (1979)

    Google Scholar 

  4. Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions results in L2(Rm), m≥ 5. Duke Math. J. 47(1), 57–80 (1980)

    MATH  Google Scholar 

  5. Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in L2(R4). J. Math. Anal. Appl. 101(2), 397–422 (1984)

  6. Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3), 583–611 (1979)

    MATH  Google Scholar 

  7. Jensen, A., Nenciu, G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6), 717–754 (2001)

    Article  MATH  Google Scholar 

  8. Journé, J.-L., Soffer, A., Sogge, C.D.: Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44(5), 573–604 (1991)

    Google Scholar 

  9. Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)

    Google Scholar 

  10. Katznelson, Y.: An introduction to harmonic analysis. New York: Dover, 1968

  11. Rauch, J.: Local decay of scattering solutions to Schrödinger’s equation. Commun. Math. Phys. 61(2), 149–168 (1978)

    MATH  Google Scholar 

  12. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York-London: Academic Press [Harcourt Brace Jovanovich, Publishers], 1978

  13. Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155, 451–513 (2004)

    Article  Google Scholar 

  14. Schlop, W.: Dispersive estimates for Schrödinger operators in dimension two, preprint 2004, to appear in Comm. Math. Phys.

  15. Weder, R.: estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential. J. Funct. Anal. 170(1), 37–68 (2000)

    Google Scholar 

  16. Weder, R.: The Wk,p-continuity of the Schrödinger wave operators on the line. Commun. Math. Phys. 208(2), 507–520 (1999)

    Article  MATH  Google Scholar 

  17. Yajima, K.: The Wk,p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47(3), 551–581 (1995)

    MATH  Google Scholar 

  18. Yajima, K.: Lp-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208(1), 125–152 (1999)

    Article  MATH  Google Scholar 

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Correspondence to M. Goldberg.

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Communicated by B. Simon

Supported by the NSF grant DMS-0070538 and a Sloan fellowship.

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Goldberg, M., Schlag, W. Dispersive Estimates for Schrödinger Operators in Dimensions One and Three. Commun. Math. Phys. 251, 157–178 (2004). https://doi.org/10.1007/s00220-004-1140-5

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  • DOI: https://doi.org/10.1007/s00220-004-1140-5

Keywords

  • Time Evolution
  • Dispersive Estimate
  • Decay Assumption