 (x) + \frac{0}{i} \int_{{\mathbb{R}}}      e^{itH_0} P_j(x,{\mbox{\boldmath ${u}$}}) \, dt, \label{eqn:0-0}\\      P_j(x,{\mbox{\boldmath ${u}$}})       &=      \left( Q_{H}(x,{\mbox{\boldmath ${u}$}}) + V_{ext}(x) \right) u_j(t,x)       + \int_{{\mathbb{R}}^n}Q_{F}(x,y,{\mbox{\boldmath ${u}$}}) u_j (t,y)\, dy