Given the time dependent equation: $$\partial_t\,\hat{\psi}(p,t) = \dfrac{p^2}{2\,m}\,\hat{\psi}(p,t) + \hat{\psi}(p,t)\star{\hat{V}(p)}$$
and forcing through some kind of separation: $\hat{\psi}(p,t) = P(p)\,T(t)$
I end up getting: $$\partial_t\,T(t)\,P(p)= \dfrac{p^2}{2\,m}\,P(p)\,T(t) + (P(p)\star{\hat{V}(p)})\cdot T(t) \\[12pt] \dfrac{\partial_t\,T(t)}{T(t)} = \dfrac{p^2/(2\,m)\,P(p)+ P(p)\star \hat{V}(p)}{P(p)}$$
thus the awkward equation $$C\,P(p) = p^2/(2\,m)\,P(p)+ P(p)\star \hat{V}(p)$$
Awkward because there is nothing to be solved in the equation. Something's wrong?
Also what's the meaning of $C$. Some kind of energy?