I wonder if there is any theory about what we can call the fractional Schrödinger equation: $$ \mathrm{i}\frac{\partial \psi}{\partial t} = (\Delta)^s \psi + g(\psi^2)\psi \quad\hbox{in $\mathbb{R}^N$.} $$ It seems to me that "stationary waves" have been studied recently, but I can't find any treatment of the evolutionary equation. In particular, is this problem wellposed in any space (with an initial condition, of course)? Any reference is welcome.
yes, there is an extensive literature, going back to Fractional Schrödinger equation, by Nick Laskin, and summarized here. 

