The conditional evolution equation includes the wave function $\Psi$ on the full configuration space in the potentials terms $A$ and $B$, or more precisely in the ratios $\psi'/\psi$ and $\psi''/\psi$. We define

\begin{equation} \phi_i(t,x) = \left. \frac{\nabla^i_y\Psi(t,x,y)}{\Psi(t,x,y)} \right|_{y=Y(t)} \end{equation}

and thus have $\phi_1 = \psi'/\psi$, $\phi_2 = \psi''/\psi$ and so on. Note that this potential quantities with $i$ even are scalar valued while those with $i$ odd are 3d vector valued. To get a general evolution equation for the potentials $\phi_i$ we take the time derivative again obeying the chain rule for the time dependent Bohmian trajectory $Y(t)$ that substitutes the coordinate $y$.

\begin{equation} \partial_t \phi_i = \left. \partial_t \frac{\nabla^i_y \Psi}{\Psi} \right|_{y=Y(t)} + \dot{Y}(t) \cdot\left. \nabla_y \frac{\nabla^i_y \Psi}{\Psi} \right|_{y=Y(t)} \end{equation}

A calculation using the Schrödinger equation with external potential $V$ and which includes the general product rule at one point yields the following lengthy expression. \begin{equation} \mathrm{i}\hbar\partial_t \phi_i = -\frac{\hbar^2}{2m} \left( \Delta_x\phi_i + 2\nabla_x\phi_i \cdot \frac{\nabla_x\psi}{\psi} + \phi_{i+2}-\phi_i \phi_2\right) + \sum_{k=1}^i {i \choose k} \left.\left( \nabla_y^k V \right)\right|_{y=Y(t)} \phi_{i-k} + \mathrm{i}\hbar \dot{Y}(t) (\phi_{i+1}-\phi_i\phi_1) \end{equation}

Note that we define $\phi_0=1$. The sum index really starts at 1 (instead of 0) because the 0th term has been cancelled. This results fits to the one given in ^[1] (35-36) for $i=1$ and $i=2$. The original wave function $\Psi$ is indeed completely absent from this scheme. We see that for small $|\psi|$ the method might be unstable because of the occurence of this quantity in the denominator, a problem that does not arise in the related coupled system of $\psi,\psi',\psi''$ etc.

Substituing the formula for the Bohmian trajectory $\dot{Y}(t) = \hbar/m \Im \phi_1(t,x)|_{x=X(t)}$ shows that also the last term has a factor $\hbar^2$ in front, thus higher orders $\phi_{i+1}$ and $\phi_{i+2}$ of the potential network are generally weighted less which hints towards a possible quick convergence of the iteration scheme. It is also interesting to note that the high frequency components of the external potential $V$ get connected to the lower orders of $\phi_i$ and vice versa with $\nabla_y^i V$ appearing by itself in the evolution equation. Of course any application must cut the infinite network of potentials at one point and then considers only the remaining orders as an approximations to the problem with full complexity.