Instead of deriving further evolution equations in the form of a potential network one can directly write down a system of coupled equations for $\psi$ (the conditional evolution equation), $\psi',\psi''$ etc. This was first noted in ^[1], a numerical realization can be found in ^[2] (Appendix A). We write \begin{equation} \psi^{(i)}(t,x) = \nabla_y^i\Psi|_{y=Y(t)} \end{equation} and get \begin{equation} \mathrm{i}\hbar\partial_t \psi^{(i)} = -\frac{\hbar^2}{2m} \left( \Delta_x\psi^{(i)} + \psi^{(i+2)}\right) + \sum_{k=0}^i {i \choose k} \left.\left( \nabla_y^k V \right)\right|_{y=Y(t)} \psi^{(i-k)} + \mathrm{i}\hbar \dot{Y}(t) \psi^{(i+1)}. \end{equation}

Written down in matrix form we have the following. \begin{equation} \mathrm{i}\hbar\partial_t \left(\begin{array}{c} \psi^{(0)} \\ \psi^{(1)} \\ \psi^{(2)} \\ \psi^{(3)} \\ \vdots \end{array}\right) = \left(\begin{array}{cccccc} H_x & \mathrm{i}\hbar\dot{Y} & -\frac{\hbar^2}{2m} & 0 & 0 & \cdots \\ \nabla_y V & H_x & \mathrm{i}\hbar\dot{Y} & -\frac{\hbar^2}{2m} & 0 & \cdots \\ \nabla_y^2 V & \nabla_y V & H_x & \mathrm{i}\hbar\dot{Y} & -\frac{\hbar^2}{2m} & \cdots \\ \nabla_y^3 V & \nabla_y^2 V & \nabla_y V & H_x & \mathrm{i}\hbar\dot{Y} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right) \cdot \left(\begin{array}{c} \psi^{(0)} \\ \psi^{(1)} \\ \psi^{(2)} \\ \psi^{(3)} \\ \vdots \end{array}\right) \end{equation}

Here $H_x = -\frac{\hbar^2}{2m}\Delta_x+V$ is the usual Hamiltonian with only the $x$-coordinates present and all terms with the potential $V$ have set $y=Y(t)$ after differentiation. Now we also notice that the given scheme is just the Taylor expansion of $\Psi(t,x,y)$ at $y=Y(t)$ (assumed to exist) and all orders together would give the full wave function $\Psi$ again. For this to hold it is imperative to check under which conditions a wave function is space-analytic and, even more restrictive, allows for a convergent Taylor series with infinite radius of convergence at the location of the Bohmian trajectory $Y(t)$.