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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 well-posed in any space (with an initial condition, of course)? Any reference is welcome.

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yes, there is an extensive literature, going back to Fractional Schrödinger equation, by Nick Laskin, and summarized here.

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