Suggested by Willie Wong, I reformulate my question posed Reference request : differential equation with operator valued unknowns to a concrete case.
I wish to emphasise that I look for the references on infinite-dimensional Lindblad equation that are written for mathematicians.
Let $\mathbf H:=L^2(\mathbb R^d)$ be a Hilbert space, and denote by $\mathcal L(\mathbf H)$ the collection of linear operators (that may not be bounded) on $\mathbf H$. For every $A\in \mathcal L(\mathbf H)$, denote by $D(A)$ the domain of $A$. Consider the Lindblad equation
\begin{equation} \mathrm{d} \rho_t = -\mathrm{i}\left[-\frac 12 \Delta + V,\rho_t\right]\mathrm{d} t + \left(K\rho_tK - \frac{1}{2}K^2\rho_t-\frac{1}{2}\rho_tK^2 \right)\mathrm{d} t, \end{equation}
where $K, V$ stand for the multiplication operator defined by the functions $k, v:\mathbb R^d\to\mathbb R$, i.e. for $\varphi\in \mathbf H$, $K(\varphi)\in \mathbf H$ is defined by
$$K(\varphi)(x):=k(x)\varphi(x),$$
and $[\cdot,\cdot]$ denotes the commutator, i.e. for $A, B\in \mathcal L(\mathbf H)$, $[A,B]:=AB-BA$. We look for the unknown $\rho$ living in the so-called trace class, i.e. $\rho_t$ is Hermitian, $\rho_t\ge 0$ and $trace(\rho_t)=1$ for all $t\ge 0$.
Question : What is the known results on the wellposedness of this equation?
If $\rho_t$ can be identified by some kernel, denoted by $\rho(t,x,y)$, the above Lindblad equation rewrites as the following PDE on $\mathbb R_+\times \mathbb R^d \times\mathbb R^d$:
\begin{eqnarray*} \partial_t \rho(t,x,y) &=& -\mathrm{i}\Big(-\frac 12 (\Delta_{x}-\Delta_{y})\rho(t,x,y) +\big( v(x)-v(y)\big) \rho(t,x,y)\Big)\\ && - \frac{1}{2}\Big(k(x)- k(y) \Big)^2 \rho(t,x,y), \end{eqnarray*}
Question: Under which conditions this PDE is well posed (admit a unique solution in the trace class)?
When $k\equiv 0$, it is known that $\rho(t,x,y)=\varphi(t,x)\overline\varphi(t,y)$, where $\varphi$ solves some Schrödinger equation, see e.g. Energy estimation of density operator to von Neumann equation
