Consider the Schrödinger equation of Hartree type (HT):

$$i\partial_tu +\Delta u + (V\ast |u|^2)u=0, u(x,0)=u_0$$ with $(x,t)\in \mathbb R^d \times \mathbb R.$ where $V$ is some potential.

(1) when $V(x)= |x|^{-\gamma} , 0<\gamma<d$ ( Hartree potential) (2) when $V(x)=e^{-|x|}|x|^{-\gamma} , 0<\gamma<d$ (Yukawa potential)

There are papers in the literature with Hartree potential and Yukawa potential.

My question is: How the well-posedness results of (HT) with Hartree potential and Yukawa potential differs? Is there any special reasons, the two different potential has been considered?

My Thought: (1)It seems, if we have well-posedness results for (HT) with Hartree potential, we will most probably have the well poshness results for (HT) with Yukawa potential. (2) I do not know many papers on (HT) with Yukawa potential, any good reference?


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