I asked the following question on [MSE](https://math.stackexchange.com/questions/4638435/ibvp-for-the-linear-homogeneous-1-d-schr%c3%b6dinger-equation) some time ago, but got no answer (sorry if the question is not appropriate for MO).

Consider the following initial boundary value problem  for the linear homogeneous 1-D Schrödinger equation for a function $u(t,x)$ in the domain $\Omega=[0,T]\times[0,L]$:
$$
\begin{cases}
iu_{t}(t,x)+\Delta u(t,x)=0,\quad(t,x)\in\Omega,
\\
u(0,x)=0,\quad x\in[0,L],
\\
u(t,0)=0,~u(t,L)=0,\quad t\in[0,T].
\end{cases}
$$
Is it true that $u(x,t)=0$ is the unique weak solution in the space
$C([0,T],L^2(\Omega))$ ?

I guess this is a basic result in the theory of IBVP for evolution equations.
\
Is there a standart reference in the literature for that result ?