Breakdown of regularity of scattering for mass-subcritical NLS

We study the scattering problem for the nonlinear Schr"odinger equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p}u$ on ${\mathbb{R}}^d$, $d\ge 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $\mathrm{\Sigma }$ holds and the wave operator is well-defined on $\mathrm{\Sigma }$. We show that there exists $0<\beta <p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+\beta }$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology.