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I was wondering what is known about the solution of the Schrödinger equation $$i h \frac{\partial}{\partial t} Ψ(x, t) =- \frac{h^2}{2m}\Delta Ψ(x,t)+V(x)Ψ(x, t)$$ for $t ∈ \mathbb{R}$. What sort of conditions are put on the potential $V$ to guarantee a solution and what space does a solution lie in? I could find information about the equation $$-i h \frac{\partial}{\partial t} Ψ(x, t) = \frac{h^2}{2m}\Delta Ψ(x,t)$$ for $x\in \mathbb{R}^n$ and $t>0$, but most everything I see about the previous equation I find hard to understand. Is there some reference where such issues are dealt with in a clear manner

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Try to find the solution in the form $Ψ(x,t)=u(x)T(t)$. –  vesszabo Oct 17 '12 at 18:59
    
You might perhaps get more answers to this on the Physics Stack Exchange. If you want, you can flag your question for ♦ moderator attention and ask for it to be migrated there. –  Ilmari Karonen Oct 17 '12 at 19:01
    
@IlmariKaronen As I am a maths student and am completely unaware of Physics, I was hoping for more math solutions –  Vivek Oct 17 '12 at 19:20
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@vesszabo I was interested in the question of existence of solution, weak or otherwise, and I was hoping for some reference which puts the problem in a proper framework and suggests the most general conditions on $V$ which would ensure solution. This paper www-m3.ma.tum.de/foswiki/pub/M3/Allgemeines/CarolineLasser/… mentions in the first paragraph that the Schrödinger equation has a global solution etc, but does not mention any source. –  Vivek Oct 17 '12 at 19:51
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Thanks for the link. I will think about it and try to ask one of my colleague. (He is an expert :-) ) –  vesszabo Oct 17 '12 at 20:06
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1 Answer

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Partial answer. Substituting $\Psi(x,t)=u(x)T(t)$ we obtain $$ ih\frac{T'(t)}{T(t)}=-\frac{h^2}{2m}\cdot\frac{\Delta u(x)}{u(x)}+V(x)=K, $$ $K$ is a constant. From this $$ T(t)=c_1 \exp\left(-\frac{iKt}{h}\right), $$ where $c_1$ is arbitrary constant. For $u(x)$ we get $$ \frac{h^2}{2m} \Delta u(x)-(V(x)-K)u(x)=0. $$ Without loss of generality we may assume that $\frac{h^2}{2m}=1$. This equation is time independent and has an enormous literature.

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