Normalizeable solutions when separating the SED

[aaaa202]

Hey guys,
I recently did an exercise whose purpose was to show what we must require from the separation constant E (when separating the SED) for the solutions to be normalized. The point was that it must be real since a complex one yields a wave function of the form:
ψ(x,t) = A(x)exp(at)
And this can't be normalized for all t. But the problem is for me: It can be normalized for all finite t, and I have shown in an earlier exercise that when the wave function is normalized at one time t0 then it stays normalized in the future (I believe this is a very important property of the SED). So isn't there some kind of inconsistensy here?

 

[dextercioby]

The necessity of imaginary exponential comes from the theorem of Stone which in turn is needed to get to the generator of time translations, the Hamiltonian. The time evolution must be represented by a norm preserving operator, thus this is necessarily unitary.