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Minor (very minor...) Math Jaxing
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Daniele Tampieri
Daniele Tampieri
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I asked the following question on MSE 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 $\Omega=[0,T]\times[0,L]$: $$ \begin{cases} iu_{t}(t,x)+\Delta u(t,x)=0,\quad(t,x)\in\Omega, \\ u(0,x)=0,\quad x\in[0,L], \\ u(t,0)=0,~u(t,L)=0,\quad t\in[0,T]. \end{cases} $$$$ \begin{cases} iu_{t}(t,x)+\Delta u(t,x)=0,&(t,x)\in\Omega, \\ u(0,x)=0, &x\in[0,L], \\ u(t,0)=0,~u(t,L)=0, &t\in[0,T]. \end{cases} $$ 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 ?

I asked the following question on MSE 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 ?

I asked the following question on MSE 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 $\Omega=[0,T]\times[0,L]$: $$ \begin{cases} iu_{t}(t,x)+\Delta u(t,x)=0,&(t,x)\in\Omega, \\ u(0,x)=0, &x\in[0,L], \\ u(t,0)=0,~u(t,L)=0, &t\in[0,T]. \end{cases} $$ 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 ?

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user111
user111
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Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation

I asked the following question on MSE 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 ?