Consider the Schrödinger equation on $\mathbb R_+\times\mathbb R^n$ as follows:
$$i\partial_t\varphi(t,x)=-\frac12\Delta_x\varphi(t,x),\quad \varphi(0,x)=\varphi_0(x).$$
Denote by $\varphi$ its solution and set $\rho(t,x,y):=\varphi(t,x)\overline{\varphi(t,y)}$. Then one obtains the so-called von Neumann equation
$$i\partial_t\rho(t,x,y)=-\frac12[\Delta_x-\Delta_y]\rho(t,x,y),\quad \rho(0,x,y)=\varphi_0(x)\overline{\varphi_0(y)}.$$
I'm looking for references for the above von Neumann equation, with general initial condition $\rho(0,x,y)=\rho_0(x,y)$, concerning the energy estimation
$$\int (|\nabla \rho_x(t,x,y)|^2 + |\nabla \rho_y(t,x,y)|^2)dxdy.$$
Here I'm particularly interested in the energy estimation of such type equation (without using the Schrödinger equation) as follows :
$$i\partial_t\rho(t,x,y)=-\frac12[\Delta_x-\Delta_y]\rho(t,x,y) + V(x,y)\rho(t,x,y),\quad \rho(0,x,y)=\rho_0(x,y),$$
where $V$ is a suitable potential function.
