On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D

Vladimir Georgiev, Chunhua Li

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the scattering problem for the nonlinear Schrödinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator A(s) appearing in commutator relations. The equivalence between the operators −ΔV [Formula presented] and −Δ [Formula presented] in L2 norm sense for 0≤s<1 is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schrödinger operator −ΔV. Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.

Original languageEnglish
Pages (from-to)208-218
Number of pages11
JournalPhysica D: Nonlinear Phenomena
Volume398
DOIs
Publication statusPublished - 2019 Nov

Keywords

  • Nonlinear Schrödinger equation
  • Resolvent estimates
  • Scattering problem
  • Strichartz estimates
  • Time decay estimates

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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