Endpoint Mapping Properties of Wave Operators for Schr\"odinger Operators in Dimensions $n\ge3$

Abstract

We study endpoint mapping properties of low-energy wave operators for Schr\"odinger operators $H=-\Delta+V$ on $\mathbb R^n$, $n\ge3$. In dimension four, we prove that a zero-energy resonance prevents $L^1$ boundedness, whether or not zero is also an eigenvalue, under $|V(x)|\lesssim\langle x\rangle^{-\beta}$ with $\beta>10$. For a zero-energy eigenvalue without a resonance, we obtain a complete low-energy $L^p$ classification in every dimension $n\ge3$ under $\beta>n+4$. In particular, $L^\infty$ boundedness is equivalent to the vanishing of the zeroth, first, and harmonic second moments of $V\psi$ for every zero-energy eigenfunction $\psi$. The proof identifies the finite-rank obstruction and shows that the remaining eigenvalue correction cannot cancel its critical asymptotic profiles. The same conclusions hold for the full wave operators when the corresponding high-energy bounds are available.