the interface conditions and satisfying $W(0)=W_0$.

In the case $T=0$ the theorem is a direct consequence of Theorem \[cor020805-19\] proved below. The more general case $T\ge0$ is treated in Section \[sec10\].

Deterministic wave equation corresponding to (\[basic:sde:2a\]) {#sec3}
===============================================================

Here we consider a deterministic part of the dynamics described in (\[basic:sde:2a\]). Its corresponding energy density function will converge to the solution of . In microscopic time the evolution of the wave function is given by $$\begin{aligned}
 \label{012503-19}
&&
\frac{d}{dt}\hat\phi(t,k)=-i{\omega}(k)\hat\phi(t,k)
 -2i{\epsilon}{\gamma}_0R(k)\hat p\left(t,k\right)-i{\gamma}_1\int_{{{\mathbb T}}}\hat{ p}\left(t,k'\right)dk',
 \\
 &&
\hat\phi(0,k)= \hat\psi(k),\nonumber
 \end{aligned}$$ where $$\hat p(t,k):=\frac{1}{2i}\left[\hat\phi(t,k)-\left(\hat\phi(t,-k)\right)^\star\right].$$ In fact it is convenient to deal with the vector formulation of the equation for $$\label{Ut}
\hat\Phi(t,k)=\left[
\begin{array}{c}
\hat\phi_+(t,k)\\
\\
\hat\phi_-(t,k),
\end{array}\right],\quad \hat\Psi(k)=\left[
\begin{array}{c}
\hat\psi_+(k)\\
\\
\hat\psi_-(k)
\end{array}\right].$$ Here, we use the convention $\hat\phi_+(t,k)=\hat\phi(t,k)$ and $\hat\phi_-(t,k):=\left(\hat\phi(t,-k)\right)^\star$ and similarly for $\hat\psi_\pm(k)$. The equation then takes the form $$\begin{aligned}
 \label{basic:sde:2av}
&&
 \frac{d}{dt}\hat\Phi(t,k)={\Omega}_{\epsilon}(k) \hat\Phi(t,k) - i\gamma_1  {\frak f}{p}_0(t) 
 ,\nonumber\\
 &&
\hat\Phi(0,k)= \hat\Psi(k).
 \end{aligned}$$ Here $$\label{Omk}
{\Omega}_{\epsilon}(k):=\left[
\begin{array}{cc}
-{\gamma}_0{\epsilon}R(k)-i{\omega}(k)&{\gamma}_0{\epsilon}R(k)\\
{\gamma}_0{\epsilon}R(k)&-{\gamma}_0{\epsilon}R(k)+i{\omega}(k)
\end{array}\right]={\Omega}_0(k)-{\gamma}_0{\epsilon}R(k){\bf D}.$$ and $${\frak f}:=\left[
\begin{array}{r}
1\\
-1
\end{array}\right],\quad {\bf D}:={\frak f}^T\otimes {\frak f}=\left[
\begin{array}{rr}
1&-1\\
-1&1
\end{array}\right].$$ The momentum at $x=0$ equals $$\label{p0}
{p}_0(t)
:=\frac{1}{2i}\int_{{{\mathbb T}}}\hat\Phi(t,k)\cdot {\frak f}dk=\frac{1}{2i}\int_{{{\mathbb T}}}\left[\hat\phi (t,k)-\left(\hat\phi (t,-k)\right)^\star\right]dk.$$ The eigenvalues of the matrix ${\Omega}_{\epsilon}(k)$ equal $
{\lambda}_\pm(k)=-{\gamma}_0{\epsilon}R(k)\pm i{\omega}_{\epsilon}(k),
$ 