The usual treatment of the Aharonov-Bohm effect (which appeared already in Aharonov and Bohm's original paper) takes two particular local solutions of the Schrödinger equation, $\psi_1$ and $\psi_2$. Here "local" means that the domains $D_1$ of $\psi_1$ and $D_2$ of $\psi_2$ are simply connected and overlapping domains whose union is a doubly-connected set which doesn't contain the solenoid and contains a circle around the solenoid.
The particularity of these solutions is that they are in the form
$$e^{iS_k}\psi_k^0 \quad (k=1,2)$$
where $S_k$ is a real-valued function on $D_k$, and $\psi_1^0+\psi_2^0$ is a solution of the Schrödinger-equation in the absence of the magnetic flux in the solenoid. These local solutions cannot be extended to $D_1\cup D_2$, because these extensions would result in multiple-valued or discontinuous functions.
And here is a gap in the logical chain, which is usually replaced by "physical" notions, such as calling $\psi_1$ and $\psi_2$ "beams" or different states (although they are clearly not states because states are global solutions of the Schrödinger equation), or referring to some "phase picking during dragging the electron" or something like these, but such things aren't defined in the mathematical model of quantum mechanics. Once we have a mathematical model, we should be able to forget the physics behind it and do pure mathematics.
From the answers I have received so far to my questions about the A-B effect, I have come to the following conclusion. The process of obtaining a global solution of the Schrödinger equation from the local solutions $\psi_1$ and $\psi_2$ can be formulated mathematically the following statement:
There exists a global, single-valued and continuous solution $\psi$ of the Schrödinger equation for which $\psi|_{D_1\cap D_2}=(\psi_1+\psi_2)|_{D_1\cap D_2}$.
But how can this statement be proved? Is this statement true at all? If not, then what the $\int A$ "Aharonov-Bohm phase" has to do with the shift of the fringes in the A-B experiment?
