Can we have a physical interpretation for a time independent Schrodinger equation of this form? I am interested in a time independent Schrodinger equation of this form. $$F*\psi - \frac{\hbar^2}{2m} \frac{\partial^2{\psi}}{\partial{x^2}} = E\psi$$
Here the product $V\psi$ is replaced by the convolution $F*\psi$. What I want to know is that is there any such $F$ where we can assign some kind of an intuitive physical meaning to it? It may be field or quantum field or whatever exotic thing.
The product replaced by convolution enables a lot of mathematically beautiful things to happen, but to begin with I strongly need a physical interpretation.
PS : My approach is to take a mathematically beautiful/appealing objects/equations and try to make sense of them in physical applications.
 A: The Fourier transformed equation wrt. $\mathbf{x}$ is
$
\tilde{F}(\mathbf{k})\tilde{\psi}(\mathbf{k})+\frac{\hbar ^{2}}{2m}\mathbf{k}%
^{2}\tilde{\psi}(\mathbf{k})=E\tilde{\psi}(\mathbf{k}),
$
so the effect is a $\mathbf{k}$-dependent contribution to the energy $E$.
A: I don't understand why the partial derivative ? If the model is one dimensional, then one can have ordinary derivative w.r.t. x. If not, one can use Laplacian. 
In Fourier analysis and elsewhere, the basic idea behind convolution is "Superposition of Waves" in some sense, so to bring a convolution of of an operator and a function, one needs to bring both of them in equally footing, if not mathematically, at least physically, so one must specify "What F is " ? (I think F replaces potential ! but one must specify what is the status of F is ? Operator or what ?) If F belongs to the same space (or class) as the eigen function belongs, then it is "kind of" integrodifferential equation, one can attach several physical meaning to the above equation, one of them is "The wave is influenced by several superposition of wave like "Sources" and glued together via convolution". One can provide more concrete information on the basis of explicit situation.
