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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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    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.