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I am wondering why every general solution to Schrodinger equation can be built from separable solutions. In other words, I don't follow that why every solution to Schrodinger equation can be written as

$$\Psi(x,t) = \sum c_n\Psi_n(x,t)=\sum c_n\psi_n(x)\phi_n(t)$$

I know that the right hand side is a solution to Schrodinger equation but this does not mean that every solution should be of this type. I also know that separable solutions form eigenbasis of time-independent Schrodinger equation but the above fact still does not follow, as they only span the solution space of the time-independent Schrodinger equation, not the general time-dependent Schrodinger equation.

What am I missing here? Thank you for your help.

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# I About general solutions to Schrodinger equation

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