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Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue

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Abstract

Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in Lp spaces of scattering solutions e−itHP c u, where P c is the orthogonal projection onto the continuous spectral subspace of L2(R3) for H. Under suitable decay assumptions on V(x) it is shown that they satisfy the so-called Lp-Lq estimates ||e−itHP c u|| p ≤(4π|t|)−3(1/2−1/p)||u|| q for all 1≤q≤2≤p≤∞ with 1/p+1/q=1 if H has no threshold resonance and eigenvalue; and for all 3/2<q≤2≤p<3 if otherwise.

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References

  1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(4), 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, 221–240 (2000)

    Google Scholar 

  3. Bergh, J., Löfström, J.: Interpolation spaces, an introduction. Berlin-Heidelberg-New York: Springer-Verlag, 1976

  4. Cuccagna, S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54, 1110–1145 (2001)

    Article  Google Scholar 

  5. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry. Berlin: Springer-Verlag, 1987

  6. Erdogan, M.B., Schlag, W.: Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three I. Dynamics of PDE, 1(4), 359 (2004)

    Google Scholar 

  7. Goldberg, M.: Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials. To appear in Geom. and Funct. Anal.

  8. Goldberg, M., Schlag, W.: Dispersive estimates for Schrödinger operators in dimensions one and three. Commun. Math. Phys. 251, 157–178 (2004)

    Article  Google Scholar 

  9. Galtbayar, A., Jensen, A., Yajima, K.: Local time-decay of solutions to Schrödinger equation with time-periodic potentials. J. Stat. Phys. 116, 231–282 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  11. Jensen, A., Mourre, E., Perry, P.: Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. Inst. H. Poincaré Phys. Théor. 41, 207–225 (1984)

    Google Scholar 

  12. Journé, J.-L., Soffer, A., Sogge, C.D.: Decay estimates for Schrödinger operators. Commun. Pure Appl. Math. 40, 573–604 (1991)

    Google Scholar 

  13. Kato, T.: Growth properties of solutions of the reduced wave equation with a variable coefficient. Commun. Pure Appl. Math. 12, 403–425 (1959)

    Google Scholar 

  14. Kato, T.: Perturbation theory for linear operators. New York: Springer Verlag, 1966

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

    Article  Google Scholar 

  16. Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 46, 113–129 (1987)

    Google Scholar 

  17. Keel, M., Tao, T.: End point Strichartz estimates. Am. J. Math. 120, 955–980 (1998)

    Google Scholar 

  18. Lions, J.L., Magenes, E.: Problémes aux limites non homogénes et applications I. Paris: Dunod, 1968

  19. Murata, M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49, 10–56 (1982)

    Article  Google Scholar 

  20. Nier, F., Soffer, A.: Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator. J. Funct. Anal. 198, 511–535 (2003)

    Article  Google Scholar 

  21. O’Neil, R.: Convolution operators and L(p,q) spaces. Duke Math. J. 30, 129–142 (1963)

    Article  Google Scholar 

  22. Reed, M., Simon, B.: Methods of moderm mathemtical physics. Vol II, Fourier analysis, selfadjointness. New York-San Francisco-London: Academic Press, 1975

  23. Reed, M., Simon, B.: Methods of moderm mathemtical physics. Vol III, Scattering theory. New York-San Francisco-London: Academic Press, 1979

  24. Reed, M., Simon, B.: Methods of moderm mathemtical physics. Vol IV, Analysis of Operators. New York-San Francisco-London: Academic Press, 1978

  25. Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time dependent potentials. Invent. Math. 155(3), 451–513 (2004)

    Article  Google Scholar 

  26. Rodnianski, I., Schlag, W., Soffer, A.: Dispersive analysis of charge transfer models. To appear in CPAM

  27. Schlag, W.: Dispersive estimates for Schrödinger operators in dimension two. Commun. Math. Phys. 257, 87–117 (2005)

    Article  Google Scholar 

  28. Weder, R.: Lp-Lp‘ estimates for the Schrödinger equations on the line and inverse scattering for the nonliner Schrödinger equation with a potential. J. Funct. Anal. 170, 37–68 (2000)

    Article  Google Scholar 

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

    Google Scholar 

  30. Yajima, K.: The Wk,p-continuity of wave operators for Schrödinger operators III. J. Math. Sci. Univ. Tokyo 2, 311–346 (1995)

    Google Scholar 

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

    Article  Google Scholar 

  32. Yajima, K.: Time Periodic Schrödinger equations, in Topics in the theory of Schrödinger operators, ed. H. Anaki and H. Ezawa, 9–69, World Scientific (2004)

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  1. Department of Mathematics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo, 171-8588, Japan

    K. Yajima

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  1. K. Yajima
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Correspondence to K. Yajima.

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Communicated by B. Simon

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Yajima, K. Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue. Commun. Math. Phys. 259, 475–509 (2005). https://doi.org/10.1007/s00220-005-1375-9

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