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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 $Ψ(x,t)=u(x)T(t)$.2012-10-17
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    You might perhaps get more answers to this on the [Physics Stack Exchange](http://physics.stackexchange.com). If you want, you can flag your question for ♦ moderator attention and ask for it to be migrated there.2012-10-17
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    @IlmariKaronen As I am a maths student and am completely unaware of Physics, I was hoping for more math solutions2012-10-17
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    @IlmariKaronen This equation is the subject of Mathematical Physics :-) So I ask to not migrate it, because the question and the possible answer is interested me and I have not account in PSE :-)2012-10-17
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    @vesszabo Can we expect solutions other than in the form $Ψ(x,t)=u(x)T(t)$?2012-10-17
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    1) If I remember well, then the name of $V$ is potential function, and in your equation it is time independent and everyone seeks the solution in this form. 2) What did you find in the literature? Your question is too general. As a first step for example assume that $V$ is continuous. Do you want "regular" solution or distributions are also permitted?2012-10-17
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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 http://www-m3.ma.tum.de/foswiki/pub/M3/Allgemeines/CarolineLasser/fermanian_lasser_SIAM.pdf mentions in the first paragraph that the Schrödinger equation has a global solution etc, but does not mention any source.2012-10-17
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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 :-) )2012-10-17

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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.