Tokyo Journal of Mathematics

$L^1$ and $L^\infty$-boundedness of Wave Operators for Three Dimensional Schrödinger Operators with Threshold Singularities

Kenji YAJIMA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

It is known that wave operators for three dimensional Schrödinger operators $-\lap + V$ with threshold singularities are bounded in $L^p(\mathbb{R}^3)$ for $1<p<3$ in general and, for $1<p<\infty$ if and only if zero energy resonances are absent and all zero energy eigenfunctions $\phi$ of $-\lap + V$ satisfy $\int V(x)x^\alpha \phi(x) dx=0$ for $|\alpha|\leq 1$. We prove here that they are bounded in $L^1(\mathbb{R}^3)$ if and only if zero energy resonances are absent. We also show that they are bounded in $L^\infty(\mathbb{R}^3)$ if no resonances are present and all zero energy eigenfunctions $\phi(x)$ satisfy $\int_{\mathbb{R}^3} x^\alpha V(x)\phi(x)dx=0$ for $0\leq |\alpha|\leq 2$. This fills the unknown parts of the $L^p$-boundedness problem for wave operators of three dimensional Schrödinger operators.

Article information

Source
Tokyo J. Math., Volume 41, Number 2 (2018), 385-406.

Dates
First available in Project Euclid: 6 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1520305215

Mathematical Reviews number (MathSciNet)
MR3908801

Zentralblatt MATH identifier
07053483

Citation

YAJIMA, Kenji. $L^1$ and $L^\infty$-boundedness of Wave Operators for Three Dimensional Schrödinger Operators with Threshold Singularities. Tokyo J. Math. 41 (2018), no. 2, 385--406. https://projecteuclid.org/euclid.tjm/1520305215


Export citation

References

  • S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 151–218.
  • G. Artbazar and K. Yajima, The $L^p$-continuity of wave operators for one dimensional Schrödinger operators, J. Math. Sci. Univ. Tokyo 7 (2000), 221–240.
  • M. Beceanu, Structure of wave operators for a scaling-critical class of potentials, Amer. J. Math. 136 (2014), 255–308.
  • J. Bergh and J. Löfström, Interpolation spaces, an introduction, Springer Verlag, Berlin-Heidelberg-New York (1976).
  • P. D'Ancona and L. Fanelli, $L^p$-boundedness of the wave operator for the one dimensional Schrödinger operator, Commun. Math. Phys. 268 (2006), 415–438.
  • D. Finco and K. Yajima, The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities. II. Even dimensional case, J. Math. Sci. Univ. Tokyo 13 (2006), 277–346.
  • A. Galtbayer and K. Yajima, Resolvent estimates in amalgam spaces and asymptotic expansions for Schrödinger equations, J. Math. Soc. Japan 65 (2013), 563–605.
  • M. Goldberg and W. Green, The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities, Advances in Math. 305 (2016), 360–389.
  • M. Goldberg and W. Green, The $L^p$ boundedness of wave operators for four dimensional Schrödinger operators with threshold singularities, arXiv:1606.06691.
  • L. Grafakos, Modern Fourier analysis, Springer Verlag, New York (2009).
  • A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979), 583–611.
  • A. Jensen and K. Yajima, A remark on $L^2$-boundedness of wave operators for two dimensional Schrödinger operators, Commun. Math. Phys. 225 (2002), 633–637.
  • A. Jensen and K. Yajima, On $L^p$-boundedness of wave operators for $4$-dimensional Schrödinger with threshold singularities, Proc. Lond. Math. Soc. (3) 96 (2008), 136–162.
  • T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403–425.
  • T. Kato, Wave operators and similarity for non-selfadjoint operators, Ann. Math. 162 (1966), 258–279.
  • S. T. Kuroda, Introduction to Scattering Theory, Lecture Notes, Matematisk Institut, Aarhus University (1978).
  • E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ. (1970).
  • R. Weder, $L^p$–$L^{p'}$ estimates for the Schrödinger equations on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal. 170 (2000), 37–68.
  • K. Yajima, The $W^{k,p}$-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan 47 (1995), 551–581.
  • K. Yajima, The $W^{k,p}$-continuity of wave operators for Schrödinger operators III, J. Math. Sci. Univ. Tokyo 2 (1995), 311–346.
  • K. Yajima, The $L^{p}$-boundedness of wave operators for two dimensional Schrödinger operators, Commun. Math. Phys. 208 (1999), 125–152.
  • K. Yajima, The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities I, Odd dimensional case, J. Math. Sci. Univ. Tokyo 7 (2006), 43–93.
  • K. Yajima, Remark on the $L^p$-boundedness of wave operators for Schrödinger operators with threshold singularities, Documenta Mathematica 21 (2016), 433–485.