$L^p$ bounds for wave operators with critical electromagnetic potentials

Abstract

We study the M\o ller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside $\frac12\Z$, we prove that the wave operators relative to $-\Delta$ exist, are unitary on $L^2$, and, together with their adjoints, are bounded on every $L^p$, $1<p<\infty$. We then specialize to the the Aharonov--Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted $L^p(\R^2,|x|^\beta\,dx)$ spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all $L^{p}$ spaces $1<p<\infty$, while boundedness fails at $p=1,\infty$. In the Krein case, both wave operators and adjoints are bounded precisely when $2/(2-\eta_\alpha)<p<2/\eta_\alpha$, where $\eta_\alpha=\max\{\alpha,1-\alpha\}$ (here $\alpha\in(0,1)$). Thus the boundary condition changes the admissible exponents.