On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Abstract

Let $H=-\Delta +V$ be a Schrödinger operator on $L^2({\mathbb{R}}^2)$ 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 $L^p({\mathbb{R}}^2)$ for all $1<p<\infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p({\mathbb{R}}^2)$ for $1<p<\infty$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents $p$.

Revised according to referee's comments. 22 pages, to appear in J. Funct. Anal

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Cited by in corpus (5)

• Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators
• On pointwise decay of waves
• On the $L^p$ boundedness of the Wave Operators for fourth order Schrödinger operators
• $L^p$-boundedness of wave operators for 2D Schrödinger operators with point interactions
• Scattering operator and wave operators for 2D Schrödinger operators with threshold obstructions