Must the wavefunction be (real) analytic? In order to show the preservation of normalization of the wave function (in one dimension for now), one shows that the time differential is zero, which entails the following step:
$$
\frac{d}{dt}\int_{-\infty}^\infty\Psi^*\Psi dx = \int_{-\infty}^\infty \frac{\partial}{\partial t}\Psi^*\Psi dx
$$
I have seen a proof of this for $\Psi = \Psi(z,t), z \in \mathbb{C}$, with the condition that $\Psi$ is analytic on some simply connected complex domain containing the limits of integration. Does this then imply that a valid wave function must have an analytic continuation to a domain which contains the real line, or is there an alternate theorem for real functions of real variables ($\Psi^*\Psi$, in this case) with weaker conditions?
 A: Let $\mathscr{H}$ be a separable Hilbert space. Suppose that the Hamiltonian $H$ is a densely defined self-adjoint operator with domain $D(H)\subset \mathscr{H}$. Then for any $\phi\in D(H)$, $i\partial_t e^{-it H}\phi=He^{-itH}\phi$, where $e^{-itH}\phi$ is the unique solution of the Schrödinger equation.
Now, $e^{-itH}$ is a unitary operator for any $t\in\mathbb{R}$. So let $\psi\in \mathscr{H}$, and consider $$\lVert e^{-itH}\psi\rVert^2=\langle e^{-itH}\psi,e^{-itH}\psi\rangle=\langle\psi,\psi\rangle=\lVert\psi\rVert^2\; ,$$
this is true for any $t\in\mathbb{R}$ and any $\psi\in\mathscr{H}$ (a general Hilbert space). So you can see that the norm is preserved by time evolution. If you really want to take the derivative in time, then you need to restrict to $\psi\in D(H)$, where the derivation makes sense as I written above. So for any $\phi\in D(H)$:
$$\frac{1}{2}\partial_t\lVert e^{-itH}\phi\rVert^2=\mathrm{Im}\langle e^{-itH}\phi,He^{-itH}\phi\rangle=0\Rightarrow \lVert e^{-itH}\phi\rVert^2=\lVert\phi\rVert^2\; .$$
The last equality can be extended to any $\psi\in \mathscr{H}$ by a density argument, using a sequence $(\psi_j)_{j\in\mathbb{N}}$ of vectors in $D(H)$ approximating $\psi$.
P.S. It is unusual to consider a wavefunction space of complex variables, but it is possible on the Bargmann-Segal representation. Anyways, you can exchange the derivation and integration signs if you can apply the dominated convergence theorem (writing the derivative as a limit), e.g. if the wavefunction is differentiable with bounded derivative. Also in the proof I wrote above, a limit procedure is implicit and justified by the differentiability in $t$ of $e^{-itH}$ (in the strong/weak operator sense in the domain of $H$/$H^{1/2}$).
