$L^p$-boundedness of wave operators for 2D Schrödinger operators with point interactions

Abstract

For two dimensional Schrödinger operator $H$ with point interactions, We prove that wave operators of scattering for the pair $(H,H_0)$, $H_0$ being the free Schrödinger operator, are bounded in the Lebesgue space $L^p({\mathbb{R}}^2)$ for $1<p<\infty$ if and only if there are no generalized eigenfunctions of $Hu(x)=0$ which satisfy $u(x)=C|x{|}^{-1}+o(|x{|}^{-1})$ as $|x|\to \infty$, $C\ne 0$. Otherwise they are bounded for $1<p\le 2$ and unbounded for $2<p<\infty$.

arXiv comment41 pages

References in corpus (5)

• Dispersive estimates for Schroedinger operators in dimension two
• L^p boundedness of the wave operator for the one dimensional Schroedinger operator
• The $L^p$ boundedness of wave operators for Schrödinger Operators with threshold singularities
• Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators
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Cited by in corpus (3)

• The $L^p$-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities
• On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction
• Scattering operator and wave operators for 2D Schrödinger operators with threshold obstructions