Abstract
We prove L1(ℝ2)→L∞(ℝ2) for the two-dimensional Schrödinger operator −Δ+V with the decay rate t−1. We assume that zero energy is neither an eigenvalue nor a resonance. This condition is formulated as in the recent paper by Jensen and Nenciu on threshold expansions for the two-dimensional resolvent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2, no. 2, 151–218 (1975)
Goldberg, M., Schlag, W.: Dispersive estimates for Schrödinger operators in dimensions one and three. Commun. Math. Phys. 251, no. 1, 157–158 (2004)
Jensen, A., Nenciu, G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13, no. 6, 717–754 (2001)
Jensen, A., Yajima, K.: A remark on Lp-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 225, no. 3, 633–637 (2002)
Journé, J.-L., Soffer, A., Sogge, C.D.: Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44, no. 5, 573–604 (1991)
Murata, M.: Asymptotic expansions in time for solutions of Schrödinger-type equations. J. Funct. Anal. 49 (1), 10–56 (1982)
Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155, 451–513 (2004)
Stoiciu, M.: An estimate for the number of bound states of the Schrödinger operator in two dimensions. Proc. Amer. Math. Soc. 132, no. 4, 1143–1151 (2004)
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, no. 1, 37–68 (2000)Yajima, K.: Lp-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208, no. 1, 125–152 (1999)
estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential. J. Funct. Anal. 170, no. 1, 37–68 (2000)Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B. Simon
The author was partially supported by the NSF grant DMS-0300081 and a Sloan Fellowship
Rights and permissions
About this article
Cite this article
Schlag, W. Dispersive Estimates for Schrödinger Operators in Dimension Two. Commun. Math. Phys. 257, 87–117 (2005). https://doi.org/10.1007/s00220-004-1262-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-004-1262-9