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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 suspect this is a basic result in the theory of IBVP for evolution equations, but I have difficulty in finding a standart reference in the literature.

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  • $\begingroup$ This is a good question! $\endgroup$
    – K.defaoite
    Feb 13 at 18:43
  • $\begingroup$ Maybe you can adapt the solution given by Tychonoff for the heat equation with zero datum? See here, here and here for more. $\endgroup$
    – Matthew Cassell
    Feb 14 at 6:34

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