On the $L^p$ boundedness of wave operators for four-dimensional Schrödinger Operators with a threshold eigenvalue

Abstract

Let $H=-\Delta +V$ be a Schrödinger operator on $L^2({\mathbb{R}}^4)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm }=s-{\lim }_{t\to \pm \infty }{e}^{itH}{e}^{-it{H}_0}$ are known to be bounded on $L^p({\mathbb{R}}^4)$ for all $1\le p\le \infty$ if zero is not an eigenvalue or resonance, and on $\frac{4}{3}<p<4$ if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on $L^p({\mathbb{R}}^4)$ for $1\le p\le \frac{4}{3}$ by direct examination of the integral kernel of the leading terms. Furthermore, if ${\int }_{{\mathbb{R}}^4}xV(x)\psi (x)dx=0$ for all zero energy eigenfunctions $\psi$, then the wave operators are bounded on $L^p$ for $1\le p<\infty$.

Updated references and made changes according to referee suggestions. To appear in Annales Henri Poincare, 20 pages

Cited by in corpus (2)

• On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions
• On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators