$L^p$-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 $2<p<\infty$.

41 pages

Cited by in corpus (1)

• On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction