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.
answered Apr 2, 2013 at 20:33
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$\begingroup$ Keep in mind that in physics literature non-local objects are sometimes treated as if they were local ones, which leads to erroneous results. For example, Laskin's solution to the "infinite potential well" problem is completely wrong (and even the Wikipedia article linked above reproduces this error). $\endgroup$ Commented Aug 23, 2017 at 22:16