Breakdown of regularity of scattering for mass-subcritical NLS

We study the scattering problem for the nonlinear Schr\"odinger equation $i\partial_t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, 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 $\Sigma$ holds and the wave operator is well-defined on $\Sigma$. 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.