Analyticity and infinite breakdown of regularity in mass-subcritical Hartree scattering

Abstract

We study the asymptotic behavior of solutions to the defocusing mass-subcritical Hartree NLS $i{u}_t+\Delta u=F(u)=(|x{|}^{-\gamma }\ast |u{|}^2)u$ on ${\mathbb{R}}^d$, $d\ge 2$, $\frac{4}{3}<\gamma <2$. We show that the scattering problem associated to this equation is analytically well-posed in the weighted spaces $\Sigma =H^1\cap \mathcal{F}{H}^1$ and $\mathcal{F}{H}^1$. Furthermore, we show that the same problem fails to be analytically well-posed for data in $L^2$. This constitutes an infinite loss of regularity between the scattering problems in weighted spaces and in $L^2$. This further develops an earlier investigation initiated by the author in which a finite breakdown of regularity was proved for the $L^2$ scattering problem for the mass-subcritical NLS with power nonlinearity $F(u)=|u{|}^{p}u$.

References in corpus (1)

• Analyticity of the scattering operator for semilinear dispersive equations