I am a mathematician studying the dynamics of the $N$-Body density matrix $\rho\_{N}(x;y)$ for $n$ particles, defined by


$$\rho\_{N, t}^{(n)} (x\_1,..,n\_n; y\_1,...,y\_n) =
\begin{cases}
\int \rho\_{N,t}(x\_1,...,x\_n,x\_{n+1},...,x\_N; y\_1,...,y\_n,x\_{n+1},...,x\_N) dx\_{n+1}...d\_{x\_N},\,\,\,\,\, 1 \leq n \le N\\
0,\,\,\,\,\, o.w.
\end{cases}$$


and the Wigner transform for one particle is:


$$w\_N(x;v):= \frac{1}{(3 \pi)^{3 N}} \int e^{- i v. y} \rho\_N \left( x + \frac{y}{2}, x- \frac{y}{2}\right) dy.$$


I cannot understand how to obtain that


$$\int w\_N(x,v) dx dv =1.$$



Integration of Wigner transform
I think your coefficients have typo's, but in any case, you first want to make sure that your reduced density matrix is properly normalized to unit trace. For one particle, that would mean that
$$\int\rho^{(1)}\_N(x,x)\,dx=1.$$
The Wigner transform can be then be defined as
$$w\_N^{(1)}(x,v)= \frac{1}{2\pi} \int e^{- i v y} \rho^{(1)}\_N \left( x + \frac{y}{2}, x- \frac{y}{2}\right)\, dy.$$
Now check that
$$\int w^{(1)}\_N(x,v) \,dx dv =\int \delta(y) \rho^{(1)}\_N \left( x + \frac{y}{2}, x- \frac{y}{2}\right)\, dxdy$$
$$\qquad=\int\rho^{(1)}\_N(x,x)\,dx=1,$$
where I used that $\int e^{-ivy}\,dy=2\pi\delta(y)$.


