linked to the time and space resolution ($\Delta x$ and $\Delta\tau$ respectively) of the simulation via the Courant condition: $$\frac{\Delta x}{\Delta\tau} > c$$ and the condition on the stability of the discretized [Schrödinger]{}equation: $$\frac{\Delta x^2}{\Delta\tau} > \hbar$$ The number of wavefunctions $N$ is chosen depending on the number of relevant modes of the decomposition in wavefunctions of the initial distribution function. The optimal value of $N$ is problem dependent and is also influenced by the algorithm chosen to discretize the distribution function. The details of this procedure will be given in section \[sec:IC\]. The precision of the original accuracy is also dictated by the choice of $\hbar$. The “quantum” nature of the formalism imposes limitations on the precision of the description of position and velocity at the same point following the equivalent of Heisenberg’s uncertainty principle.

Take an initial phase-space distribution function of the matter fields $f(x,v)$ (in the case of cosmological simulations, it is expressed via the power spectrum $P(k)$). Decompose the distribution function in $N$ complex-valued $\psi_n(x)$ such that $$\label{eq:initial_distribution}
f(x,v) \approx \sum_{n=1}^N  \int 
e^{\frac{i}{\hbar}v\,y}\psi_n^*\left(x+\frac{y}{2}\right)\psi_n\left(x-\frac{y}{2}\right)
d^3y.$$ The number $N$ of wavefunctions is chosen such as to minimize the error introduced by the decomposition and will, in practice, be 