Wikipedia about the prediction of the Aharonov–Bohm effect:
Werner Ehrenberg (1901–1975) and Raymond E. Siday first predicted the effect in 1949.
What was the theoretical basis for this prediction?
And BTW, an older question about Has The Aharonov-Bohm Effect Been Experimentally Proven? is not answered clearly. It seems that the claimed Aharonov–Bohm effect in 1959 was a prediction too. So which different theoretical basis was used by Aharonov and Bohm?
It's well described in the wikipedia page, but it just falls out from the Hamiltonian for a particle in a region with non-zero $\vec{A}$ but not necessarily non-zero $\vec{B}$ or $\vec{E}$:
Using cylindrical coordinates for a particle confined to a ring of radius $R$ with a solenoid of radius $r < R$ inside,
$H = \frac{1}{2m} \left( -\hbar^2\nabla^2 + \frac{2i\hbar q}{c}\vec{A}\cdot \vec{\nabla} + \frac{q^2}{c^2}|\vec{A}|^2 \right)$
This can be expressed
$H = -\frac{\hbar^2}{2mR^2} \left( \frac{\partial}{\partial \theta} - K \right)^2; K = i\frac{q \Phi}{2c\hbar}$
Where $\Phi$ is the flux. When you work out solutions with boundary conditions $\psi(\theta) = \psi(\theta + 2\pi)$, you will see the energy only cares about the flux, but the wavefunction itself still depends on $\vec{A}$, lending itself to potential interference effects, hence the Abraham Bohm effect.
